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Theorem fpwwe2 9450
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 8838. 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 9447 . . . . . . . . . 10 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6 ffun 6035 . . . . . . . . . 10 (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊)
75, 6syl 17 . . . . . . . . 9 (𝜑 → Fun 𝑊)
8 funbrfv2b 6227 . . . . . . . . 9 (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
97, 8syl 17 . . . . . . . 8 (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
109simprbda 652 . . . . . . 7 ((𝜑𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
1110adantrr 752 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
12 elssuni 4458 . . . . . . 7 (𝑌 ∈ dom 𝑊𝑌 dom 𝑊)
1312, 4syl6sseqr 3644 . . . . . 6 (𝑌 ∈ dom 𝑊𝑌𝑋)
1411, 13syl 17 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑋)
15 simpl 473 . . . . . . 7 ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌)
1615a1i 11 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌))
17 simplrr 800 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
182adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ V)
1918adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ V)
201, 2, 3, 4fpwwe2lem12 9448 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ dom 𝑊)
21 funfvbrb 6316 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑊 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
227, 21syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
2320, 22mpbid 222 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑊(𝑊𝑋))
241, 2fpwwe2lem2 9439 . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)))
2827simpld 475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝐴)
2919, 28ssexd 4796 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
30 difexg 4799 . . . . . . . . . . . . 13 (𝑋 ∈ V → (𝑋𝑌) ∈ V)
3129, 30syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ∈ V)
3226simprd 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
3332simpld 475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) We 𝑋)
34 wefr 5094 . . . . . . . . . . . . 13 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Fr 𝑋)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) Fr 𝑋)
36 difssd 3730 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ⊆ 𝑋)
37 fri 5066 . . . . . . . . . . . . 13 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ ((𝑋𝑌) ⊆ 𝑋 ∧ (𝑋𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
3837expr 642 . . . . . . . . . . . 12 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ (𝑋𝑌) ⊆ 𝑋) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
3931, 35, 36, 38syl21anc 1323 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
40 ssdif0 3933 . . . . . . . . . . . . . . 15 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41 indif1 3863 . . . . . . . . . . . . . . . 16 ((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌)
4241eqeq1i 2625 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
43 disj 4008 . . . . . . . . . . . . . . . 16 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
44 vex 3198 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
45 vex 3198 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ V
4645eliniseg 5482 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ V → (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧))
4744, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧)
4847notbii 310 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊𝑋)𝑧)
4948ralbii 2977 . . . . . . . . . . . . . . . 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 5473 . . . . . . . . . . . . . . . . 17 ((𝑊𝑋) “ {𝑧}) ⊆ dom (𝑊𝑋)
5327simprd 479 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) ⊆ (𝑋 × 𝑋))
54 dmss 5312 . . . . . . . . . . . . . . . . . . 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 5334 . . . . . . . . . . . . . . . . . 18 dom (𝑋 × 𝑋) = 𝑋
5755, 56syl6sseq 3643 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ 𝑋)
5852, 57syl5ss 3606 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑋)
59 sseqin2 3809 . . . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . . 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 3048 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
64 eldifn 3725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑋𝑌) → ¬ 𝑧𝑌)
6564ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧𝑌)
66 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑌𝑧𝑌))
6766notbid 308 . . . . . . . . . . . . . . . . . . . . . . . 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 129 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌 → ¬ 𝑤 = 𝑧))
7069imp 445 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑤 = 𝑧)
7165adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧𝑌)
72 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7372ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7473breqd 4655 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤))
75 eldifi 3724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (𝑋𝑌) → 𝑧𝑋)
7675ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧𝑋)
7776adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧𝑋)
78 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑌)
79 brxp 5137 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧𝑋𝑤𝑌))
8077, 78, 79sylanbrc 697 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
81 brin 4695 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊𝑋)𝑤𝑧(𝑋 × 𝑌)𝑤))
8281rbaib 946 . . . . . . . . . . . . . . . . . . . . . . . . 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 9439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8685biimpa 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑌𝑊𝑅) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8786adantrr 752 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8887simpld 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)))
8988simprd 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
9089ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
9190ssbrd 4687 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑌 × 𝑌)𝑤))
92 brxp 5137 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧𝑌𝑤𝑌))
9392simplbi 476 . . . . . . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) We 𝑋)
98 weso 5095 . . . . . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌𝑋)
101100sselda 3595 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑋)
102 sotric 5051 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤)))
103 ioran 511 . . . . . . . . . . . . . . . . . . . . . . 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 1322 . . . . . . . . . . . . . . . . . . . . 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 956 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌𝑤 ∈ ((𝑊𝑋) “ {𝑧})))
109108ssrdv 3601 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ ((𝑊𝑋) “ {𝑧}))
110 simprr 795 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)
111109, 110eqssd 3612 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = ((𝑊𝑋) “ {𝑧}))
112 in32 3817 . . . . . . . . . . . . . . . . . 18 (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
113 simplrr 800 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
114113ineq1d 3805 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
11589ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
116 df-ss 3581 . . . . . . . . . . . . . . . . . . . 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 2656 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
119 inss2 3826 . . . . . . . . . . . . . . . . . . . 20 ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
120 xpss1 5218 . . . . . . . . . . . . . . . . . . . . 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 3606 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
123 df-ss 3581 . . . . . . . . . . . . . . . . . . 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 2678 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
126111sqxpeqd 5131 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))
127126ineq2d 3806 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
128125, 127eqtrd 2654 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
129111, 128oveq12d 6653 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))))
13019adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ V)
13123adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊𝑋))
132131ad2antrr 761 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊𝑋))
1331, 130, 132fpwwe2lem3 9440 . . . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . . . 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 2654 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
136135, 65eqneltrd 2718 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
137136rexlimdvaa 3028 . . . . . . . . . . . 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 2810 . . . . . . . . 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 3933 . . . . . . . 8 (𝑋𝑌 ↔ (𝑋𝑌) = ∅)
143141, 142sylibr 224 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝑌)
144143ex 450 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌))) → 𝑋𝑌))
1453adantlr 750 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
146 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
1471, 18, 145, 131, 146fpwwe2lem10 9446 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))))
14816, 144, 147mpjaod 396 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑌)
14914, 148eqssd 3612 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
1507adantr 481 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
151149, 146eqbrtrrd 4668 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
152 funbrfv 6221 . . . . . 6 (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊𝑋) = 𝑅))
153150, 151, 152sylc 65 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊𝑋) = 𝑅)
154153eqcomd 2626 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊𝑋))
155149, 154jca 554 . . 3 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋𝑅 = (𝑊𝑋)))
156155ex 450 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
1571, 2, 3, 4fpwwe2lem13 9449 . . . 4 (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
15823, 157jca 554 . . 3 (𝜑 → (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
159 breq12 4649 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅𝑋𝑊(𝑊𝑋)))
160 oveq12 6644 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊𝑋)))
161 simpl 473 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → 𝑌 = 𝑋)
162160, 161eleq12d 2693 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
163159, 162anbi12d 746 . . 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  wo 383  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  [wsbc 3429  cdif 3564  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149  {csn 4168   cuni 4427   class class class wbr 4644  {copab 4703   Or wor 5024   Fr wfr 5060   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  cima 5107  Fun wfun 5870  wf 5872  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-wrecs 7392  df-recs 7453  df-oi 8400
This theorem is referenced by:  fpwwe  9453  canthwelem  9457  pwfseqlem4  9469
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