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Theorem fpwwe2lem7 9496
Description: Lemma for fpwwe2 9503. (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)
fpwwe2lem9.x (𝜑𝑋𝑊𝑅)
fpwwe2lem9.y (𝜑𝑌𝑊𝑆)
fpwwe2lem9.m 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem9.n 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem7.1 (𝜑𝐵 ∈ dom 𝑀)
fpwwe2lem7.2 (𝜑𝐵 ∈ dom 𝑁)
fpwwe2lem7.3 (𝜑 → (𝑀𝐵) = (𝑁𝐵))
Assertion
Ref Expression
fpwwe2lem7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑆(𝑁𝐵) ∧ (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷))))
Distinct variable groups:   𝑦,𝑢,𝐵   𝑢,𝑟,𝑥,𝑦,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝑀,𝑟,𝑢,𝑥,𝑦   𝑁,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑆,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)   𝐵(𝑥,𝑟)   𝐶(𝑥,𝑦,𝑢,𝑟)   𝐷(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem7
StepHypRef Expression
1 fpwwe2lem9.y . . . . . . . 8 (𝜑𝑌𝑊𝑆)
2 fpwwe2.1 . . . . . . . . . 10 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
32relopabi 5278 . . . . . . . . 9 Rel 𝑊
43brrelexi 5192 . . . . . . . 8 (𝑌𝑊𝑆𝑌 ∈ V)
51, 4syl 17 . . . . . . 7 (𝜑𝑌 ∈ V)
6 fpwwe2.2 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
72, 6fpwwe2lem2 9492 . . . . . . . . . 10 (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
81, 7mpbid 222 . . . . . . . . 9 (𝜑 → ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
98simprd 478 . . . . . . . 8 (𝜑 → (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))
109simpld 474 . . . . . . 7 (𝜑𝑆 We 𝑌)
11 fpwwe2lem9.n . . . . . . . 8 𝑁 = OrdIso(𝑆, 𝑌)
1211oiiso 8483 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
135, 10, 12syl2anc 694 . . . . . 6 (𝜑𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
1413adantr 480 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
15 isof1o 6613 . . . . 5 (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁1-1-onto𝑌)
1614, 15syl 17 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁:dom 𝑁1-1-onto𝑌)
17 fpwwe2.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
18 fpwwe2lem9.x . . . . . 6 (𝜑𝑋𝑊𝑅)
19 fpwwe2lem9.m . . . . . 6 𝑀 = OrdIso(𝑅, 𝑋)
20 fpwwe2lem7.1 . . . . . 6 (𝜑𝐵 ∈ dom 𝑀)
21 fpwwe2lem7.2 . . . . . 6 (𝜑𝐵 ∈ dom 𝑁)
22 fpwwe2lem7.3 . . . . . 6 (𝜑 → (𝑀𝐵) = (𝑁𝐵))
232, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 9495 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))
2423simp2d 1094 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑌)
25 f1ocnvfv2 6573 . . . 4 ((𝑁:dom 𝑁1-1-onto𝑌𝐶𝑌) → (𝑁‘(𝑁𝐶)) = 𝐶)
2616, 24, 25syl2anc 694 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁‘(𝑁𝐶)) = 𝐶)
2723simp3d 1095 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) = (𝑁𝐶))
283brrelexi 5192 . . . . . . . . . . . 12 (𝑋𝑊𝑅𝑋 ∈ V)
2918, 28syl 17 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
302, 6fpwwe2lem2 9492 . . . . . . . . . . . . . 14 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
3118, 30mpbid 222 . . . . . . . . . . . . 13 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
3231simprd 478 . . . . . . . . . . . 12 (𝜑 → (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3332simpld 474 . . . . . . . . . . 11 (𝜑𝑅 We 𝑋)
3419oiiso 8483 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3529, 33, 34syl2anc 694 . . . . . . . . . 10 (𝜑𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3635adantr 480 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
37 isof1o 6613 . . . . . . . . 9 (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀1-1-onto𝑋)
3836, 37syl 17 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀:dom 𝑀1-1-onto𝑋)
3923simp1d 1093 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑋)
40 f1ocnvfv2 6573 . . . . . . . 8 ((𝑀:dom 𝑀1-1-onto𝑋𝐶𝑋) → (𝑀‘(𝑀𝐶)) = 𝐶)
4138, 39, 40syl2anc 694 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶)) = 𝐶)
42 simpr 476 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑅(𝑀𝐵))
4341, 42eqbrtrd 4707 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵))
44 f1ocnv 6187 . . . . . . . . 9 (𝑀:dom 𝑀1-1-onto𝑋𝑀:𝑋1-1-onto→dom 𝑀)
45 f1of 6175 . . . . . . . . 9 (𝑀:𝑋1-1-onto→dom 𝑀𝑀:𝑋⟶dom 𝑀)
4638, 44, 453syl 18 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀:𝑋⟶dom 𝑀)
4746, 39ffvelrnd 6400 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) ∈ dom 𝑀)
4820adantr 480 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐵 ∈ dom 𝑀)
49 isorel 6616 . . . . . . 7 ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((𝑀𝐶) ∈ dom 𝑀𝐵 ∈ dom 𝑀)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5036, 47, 48, 49syl12anc 1364 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5143, 50mpbird 247 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) E 𝐵)
5227, 51eqbrtrrd 4709 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐶) E 𝐵)
53 f1ocnv 6187 . . . . . . 7 (𝑁:dom 𝑁1-1-onto𝑌𝑁:𝑌1-1-onto→dom 𝑁)
54 f1of 6175 . . . . . . 7 (𝑁:𝑌1-1-onto→dom 𝑁𝑁:𝑌⟶dom 𝑁)
5516, 53, 543syl 18 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁:𝑌⟶dom 𝑁)
5655, 24ffvelrnd 6400 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐶) ∈ dom 𝑁)
5721adantr 480 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐵 ∈ dom 𝑁)
58 isorel 6616 . . . . 5 ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ ((𝑁𝐶) ∈ dom 𝑁𝐵 ∈ dom 𝑁)) → ((𝑁𝐶) E 𝐵 ↔ (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵)))
5914, 56, 57, 58syl12anc 1364 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑁𝐶) E 𝐵 ↔ (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵)))
6052, 59mpbid 222 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵))
6126, 60eqbrtrrd 4709 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑆(𝑁𝐵))
6227adantrr 753 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝑀𝐶) = (𝑁𝐶))
632, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 9495 . . . . . . 7 ((𝜑𝐷𝑅(𝑀𝐵)) → (𝐷𝑋𝐷𝑌 ∧ (𝑀𝐷) = (𝑁𝐷)))
6463simp3d 1095 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → (𝑀𝐷) = (𝑁𝐷))
6564adantrl 752 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝑀𝐷) = (𝑁𝐷))
6662, 65breq12d 4698 . . . 4 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → ((𝑀𝐶) E (𝑀𝐷) ↔ (𝑁𝐶) E (𝑁𝐷)))
6735adantr 480 . . . . . 6 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
68 isocnv 6620 . . . . . 6 (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
6967, 68syl 17 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
7039adantrr 753 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐶𝑋)
7131simpld 474 . . . . . . . . . 10 (𝜑 → (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)))
7271simprd 478 . . . . . . . . 9 (𝜑𝑅 ⊆ (𝑋 × 𝑋))
7372ssbrd 4728 . . . . . . . 8 (𝜑 → (𝐷𝑅(𝑀𝐵) → 𝐷(𝑋 × 𝑋)(𝑀𝐵)))
7473imp 444 . . . . . . 7 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷(𝑋 × 𝑋)(𝑀𝐵))
75 brxp 5181 . . . . . . . 8 (𝐷(𝑋 × 𝑋)(𝑀𝐵) ↔ (𝐷𝑋 ∧ (𝑀𝐵) ∈ 𝑋))
7675simplbi 475 . . . . . . 7 (𝐷(𝑋 × 𝑋)(𝑀𝐵) → 𝐷𝑋)
7774, 76syl 17 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷𝑋)
7877adantrl 752 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐷𝑋)
79 isorel 6616 . . . . 5 ((𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝑅𝐷 ↔ (𝑀𝐶) E (𝑀𝐷)))
8069, 70, 78, 79syl12anc 1364 . . . 4 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝐶𝑅𝐷 ↔ (𝑀𝐶) E (𝑀𝐷)))
8113adantr 480 . . . . . 6 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
82 isocnv 6620 . . . . . 6 (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
8381, 82syl 17 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
8424adantrr 753 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐶𝑌)
8563simp2d 1094 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷𝑌)
8685adantrl 752 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐷𝑌)
87 isorel 6616 . . . . 5 ((𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝐶𝑌𝐷𝑌)) → (𝐶𝑆𝐷 ↔ (𝑁𝐶) E (𝑁𝐷)))
8883, 84, 86, 87syl12anc 1364 . . . 4 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝐶𝑆𝐷 ↔ (𝑁𝐶) E (𝑁𝐷)))
8966, 80, 883bitr4d 300 . . 3 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝐶𝑅𝐷𝐶𝑆𝐷))
9089expr 642 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷)))
9161, 90jca 553 1 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑆(𝑁𝐵) ∧ (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cin 3606  wss 3607  {csn 4210   class class class wbr 4685  {copab 4745   E cep 5057   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  wf 5922  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  OrdIsocoi 8455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-wrecs 7452  df-recs 7513  df-oi 8456
This theorem is referenced by:  fpwwe2lem8  9497
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