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Theorem fpwwe2 9666
Description: Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9052. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)

Proof of Theorem fpwwe2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2.1 . . . . . . . . . . 11 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2 fpwwe2.2 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
3 fpwwe2.3 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4 fpwwe2.4 . . . . . . . . . . 11 𝑋 = dom 𝑊
51, 2, 3, 4fpwwe2lem11 9663 . . . . . . . . . 10 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6 ffun 6188 . . . . . . . . . 10 (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊)
75, 6syl 17 . . . . . . . . 9 (𝜑 → Fun 𝑊)
8 funbrfv2b 6382 . . . . . . . . 9 (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
97, 8syl 17 . . . . . . . 8 (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
109simprbda 480 . . . . . . 7 ((𝜑𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
1110adantrr 688 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
12 elssuni 4601 . . . . . . 7 (𝑌 ∈ dom 𝑊𝑌 dom 𝑊)
1312, 4syl6sseqr 3799 . . . . . 6 (𝑌 ∈ dom 𝑊𝑌𝑋)
1411, 13syl 17 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑋)
15 simpl 468 . . . . . . 7 ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌)
1615a1i 11 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌))
17 simplrr 755 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
182adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ V)
1918adantr 466 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ V)
201, 2, 3, 4fpwwe2lem12 9664 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ dom 𝑊)
21 funfvbrb 6473 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑊 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
227, 21syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
2320, 22mpbid 222 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑊(𝑊𝑋))
241, 2fpwwe2lem2 9655 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋𝑊(𝑊𝑋) ↔ ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
2523, 24mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2625ad2antrr 697 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2726simpld 476 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)))
2827simpld 476 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝐴)
2919, 28ssexd 4936 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
30 difexg 4939 . . . . . . . . . . . . 13 (𝑋 ∈ V → (𝑋𝑌) ∈ V)
3129, 30syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ∈ V)
3226simprd 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
3332simpld 476 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) We 𝑋)
34 wefr 5239 . . . . . . . . . . . . 13 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Fr 𝑋)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) Fr 𝑋)
36 difssd 3887 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ⊆ 𝑋)
37 fri 5211 . . . . . . . . . . . . 13 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ ((𝑋𝑌) ⊆ 𝑋 ∧ (𝑋𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
3837expr 444 . . . . . . . . . . . 12 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ (𝑋𝑌) ⊆ 𝑋) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
3931, 35, 36, 38syl21anc 1474 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
40 ssdif0 4087 . . . . . . . . . . . . . . 15 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41 indif1 4018 . . . . . . . . . . . . . . . 16 ((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌)
4241eqeq1i 2775 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
43 disj 4158 . . . . . . . . . . . . . . . 16 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
44 vex 3352 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
45 vex 3352 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ V
4645eliniseg 5635 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ V → (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧))
4744, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧)
4847notbii 309 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊𝑋)𝑧)
4948ralbii 3128 . . . . . . . . . . . . . . . 16 (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
5043, 49bitri 264 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
5140, 42, 503bitr2i 288 . . . . . . . . . . . . . 14 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
52 cnvimass 5626 . . . . . . . . . . . . . . . . 17 ((𝑊𝑋) “ {𝑧}) ⊆ dom (𝑊𝑋)
5327simprd 477 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) ⊆ (𝑋 × 𝑋))
54 dmss 5461 . . . . . . . . . . . . . . . . . . 19 ((𝑊𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
5553, 54syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
56 dmxpid 5483 . . . . . . . . . . . . . . . . . 18 dom (𝑋 × 𝑋) = 𝑋
5755, 56syl6sseq 3798 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ 𝑋)
5852, 57syl5ss 3761 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑋)
59 sseqin2 3966 . . . . . . . . . . . . . . . 16 (((𝑊𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
6058, 59sylib 208 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
6160sseq1d 3779 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6251, 61syl5bbr 274 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6362rexbidv 3199 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
64 eldifn 3882 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑋𝑌) → ¬ 𝑧𝑌)
6564ad2antrl 699 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧𝑌)
66 eleq1w 2832 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑌𝑧𝑌))
6766notbid 307 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (¬ 𝑤𝑌 ↔ ¬ 𝑧𝑌))
6865, 67syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤𝑌))
6968con2d 131 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌 → ¬ 𝑤 = 𝑧))
7069imp 393 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑤 = 𝑧)
7165adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧𝑌)
72 simprr 748 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7372ad2antrr 697 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7473breqd 4795 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤))
75 eldifi 3881 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (𝑋𝑌) → 𝑧𝑋)
7675ad2antrl 699 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧𝑋)
7776adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧𝑋)
78 simpr 471 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑌)
79 brxp 5287 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧𝑋𝑤𝑌))
8077, 78, 79sylanbrc 564 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
81 brin 4836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊𝑋)𝑤𝑧(𝑋 × 𝑌)𝑤))
8281rbaib 520 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8380, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8474, 83bitrd 268 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑊𝑋)𝑤))
851, 2fpwwe2lem2 9655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8685biimpa 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑌𝑊𝑅) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8786adantrr 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8887simpld 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)))
8988simprd 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
9089ad3antrrr 701 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
9190ssbrd 4827 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑌 × 𝑌)𝑤))
92 brxp 5287 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧𝑌𝑤𝑌))
9392simplbi 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(𝑌 × 𝑌)𝑤𝑧𝑌)
9491, 93syl6 35 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧𝑌))
9584, 94sylbird 250 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧(𝑊𝑋)𝑤𝑧𝑌))
9671, 95mtod 189 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧(𝑊𝑋)𝑤)
9733ad2antrr 697 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) We 𝑋)
98 weso 5240 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Or 𝑋)
9997, 98syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) Or 𝑋)
10014ad2antrr 697 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌𝑋)
101100sselda 3750 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑋)
102 sotric 5196 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤)))
103 ioran 912 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤))
104102, 103syl6bb 276 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10599, 101, 77, 104syl12anc 1473 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10670, 96, 105mpbir2and 684 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤(𝑊𝑋)𝑧)
107106, 47sylibr 224 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
108107ex 397 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌𝑤 ∈ ((𝑊𝑋) “ {𝑧})))
109108ssrdv 3756 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ ((𝑊𝑋) “ {𝑧}))
110 simprr 748 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)
111109, 110eqssd 3767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = ((𝑊𝑋) “ {𝑧}))
112 in32 3972 . . . . . . . . . . . . . . . . . 18 (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
113 simplrr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
114113ineq1d 3962 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
11589ad2antrr 697 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
116 df-ss 3735 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
117115, 116sylib 208 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
118114, 117eqtr3d 2806 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
119 inss2 3980 . . . . . . . . . . . . . . . . . . . 20 ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
120 xpss1 5267 . . . . . . . . . . . . . . . . . . . . 21 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
121100, 120syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
122119, 121syl5ss 3761 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
123 df-ss 3735 . . . . . . . . . . . . . . . . . . 19 (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
124122, 123sylib 208 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
125112, 118, 1243eqtr3a 2828 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
126111sqxpeqd 5281 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))
127126ineq2d 3963 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
128125, 127eqtrd 2804 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
129111, 128oveq12d 6810 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))))
13019adantr 466 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ V)
13123adantr 466 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊𝑋))
132131ad2antrr 697 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊𝑋))
1331, 130, 132fpwwe2lem3 9656 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧𝑋) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
13476, 133mpdan 659 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
135129, 134eqtrd 2804 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
136135, 65eqneltrd 2868 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
137136rexlimdvaa 3179 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13863, 137sylbid 230 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13939, 138syld 47 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
140139necon4ad 2961 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋𝑌) = ∅))
14117, 140mpd 15 . . . . . . . 8 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) = ∅)
142 ssdif0 4087 . . . . . . . 8 (𝑋𝑌 ↔ (𝑋𝑌) = ∅)
143141, 142sylibr 224 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝑌)
144143ex 397 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌))) → 𝑋𝑌))
1453adantlr 686 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
146 simprl 746 . . . . . . 7 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
1471, 18, 145, 131, 146fpwwe2lem10 9662 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))))
14816, 144, 147mpjaod 840 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑌)
14914, 148eqssd 3767 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
1507adantr 466 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
151149, 146eqbrtrrd 4808 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
152 funbrfv 6375 . . . . . 6 (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊𝑋) = 𝑅))
153150, 151, 152sylc 65 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊𝑋) = 𝑅)
154153eqcomd 2776 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊𝑋))
155149, 154jca 495 . . 3 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋𝑅 = (𝑊𝑋)))
156155ex 397 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
1571, 2, 3, 4fpwwe2lem13 9665 . . . 4 (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
15823, 157jca 495 . . 3 (𝜑 → (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
159 breq12 4789 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅𝑋𝑊(𝑊𝑋)))
160 oveq12 6801 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊𝑋)))
161 simpl 468 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → 𝑌 = 𝑋)
162160, 161eleq12d 2843 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
163159, 162anbi12d 608 . . 3 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)))
164158, 163syl5ibrcom 237 . 2 (𝜑 → ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
165156, 164impbid 202 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  [wsbc 3585  cdif 3718  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314   cuni 4572   class class class wbr 4784  {copab 4844   Or wor 5169   Fr wfr 5205   We wwe 5207   × cxp 5247  ccnv 5248  dom cdm 5249  cima 5252  Fun wfun 6025  wf 6027  cfv 6031  (class class class)co 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-wrecs 7558  df-recs 7620  df-oi 8570
This theorem is referenced by:  fpwwe  9669  canthwelem  9673  pwfseqlem4  9685
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