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Theorem fpwwe2lem7 9337
 Description: Lemma for fpwwe2 9344. (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 5167 . . . . . . . . 9 Rel 𝑊
43brrelexi 5082 . . . . . . . 8 (𝑌𝑊𝑆𝑌 ∈ V)
51, 4syl 17 . . . . . . 7 (𝜑𝑌 ∈ V)
6 fpwwe2.2 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
72, 6fpwwe2lem2 9333 . . . . . . . . . 10 (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌𝐴𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦𝑌 [(𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
81, 7mpbid 221 . . . . . . . . 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 8325 . . . . . . 7 ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
135, 10, 12syl2anc 691 . . . . . 6 (𝜑𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
1413adantr 480 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
15 isof1o 6473 . . . . 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 9336 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))
2423simp2d 1067 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑌)
25 f1ocnvfv2 6433 . . . 4 ((𝑁:dom 𝑁1-1-onto𝑌𝐶𝑌) → (𝑁‘(𝑁𝐶)) = 𝐶)
2616, 24, 25syl2anc 691 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁‘(𝑁𝐶)) = 𝐶)
2723simp3d 1068 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) = (𝑁𝐶))
283brrelexi 5082 . . . . . . . . . . . 12 (𝑋𝑊𝑅𝑋 ∈ V)
2918, 28syl 17 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
302, 6fpwwe2lem2 9333 . . . . . . . . . . . . . 14 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
3118, 30mpbid 221 . . . . . . . . . . . . 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 8325 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3529, 33, 34syl2anc 691 . . . . . . . . . 10 (𝜑𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
3635adantr 480 . . . . . . . . 9 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
37 isof1o 6473 . . . . . . . . 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 1066 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑋)
40 f1ocnvfv2 6433 . . . . . . . 8 ((𝑀:dom 𝑀1-1-onto𝑋𝐶𝑋) → (𝑀‘(𝑀𝐶)) = 𝐶)
4138, 39, 40syl2anc 691 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶)) = 𝐶)
42 simpr 476 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑅(𝑀𝐵))
4341, 42eqbrtrd 4605 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵))
44 f1ocnv 6062 . . . . . . . . 9 (𝑀:dom 𝑀1-1-onto𝑋𝑀:𝑋1-1-onto→dom 𝑀)
45 f1of 6050 . . . . . . . . 9 (𝑀:𝑋1-1-onto→dom 𝑀𝑀:𝑋⟶dom 𝑀)
4638, 44, 453syl 18 . . . . . . . 8 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑀:𝑋⟶dom 𝑀)
4746, 39ffvelrnd 6268 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) ∈ dom 𝑀)
4820adantr 480 . . . . . . 7 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐵 ∈ dom 𝑀)
49 isorel 6476 . . . . . . 7 ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((𝑀𝐶) ∈ dom 𝑀𝐵 ∈ dom 𝑀)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5036, 47, 48, 49syl12anc 1316 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑀𝐶) E 𝐵 ↔ (𝑀‘(𝑀𝐶))𝑅(𝑀𝐵)))
5143, 50mpbird 246 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑀𝐶) E 𝐵)
5227, 51eqbrtrrd 4607 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐶) E 𝐵)
53 f1ocnv 6062 . . . . . . 7 (𝑁:dom 𝑁1-1-onto𝑌𝑁:𝑌1-1-onto→dom 𝑁)
54 f1of 6050 . . . . . . 7 (𝑁:𝑌1-1-onto→dom 𝑁𝑁:𝑌⟶dom 𝑁)
5516, 53, 543syl 18 . . . . . 6 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝑁:𝑌⟶dom 𝑁)
5655, 24ffvelrnd 6268 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁𝐶) ∈ dom 𝑁)
5721adantr 480 . . . . 5 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐵 ∈ dom 𝑁)
58 isorel 6476 . . . . 5 ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ ((𝑁𝐶) ∈ dom 𝑁𝐵 ∈ dom 𝑁)) → ((𝑁𝐶) E 𝐵 ↔ (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵)))
5914, 56, 57, 58syl12anc 1316 . . . 4 ((𝜑𝐶𝑅(𝑀𝐵)) → ((𝑁𝐶) E 𝐵 ↔ (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵)))
6052, 59mpbid 221 . . 3 ((𝜑𝐶𝑅(𝑀𝐵)) → (𝑁‘(𝑁𝐶))𝑆(𝑁𝐵))
6126, 60eqbrtrrd 4607 . 2 ((𝜑𝐶𝑅(𝑀𝐵)) → 𝐶𝑆(𝑁𝐵))
6227adantrr 749 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝑀𝐶) = (𝑁𝐶))
632, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 9336 . . . . . . 7 ((𝜑𝐷𝑅(𝑀𝐵)) → (𝐷𝑋𝐷𝑌 ∧ (𝑀𝐷) = (𝑁𝐷)))
6463simp3d 1068 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → (𝑀𝐷) = (𝑁𝐷))
6564adantrl 748 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝑀𝐷) = (𝑁𝐷))
6662, 65breq12d 4596 . . . 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 6480 . . . . . 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 749 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐶𝑋)
7131simpld 474 . . . . . . . . . 10 (𝜑 → (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)))
7271simprd 478 . . . . . . . . 9 (𝜑𝑅 ⊆ (𝑋 × 𝑋))
7372ssbrd 4626 . . . . . . . 8 (𝜑 → (𝐷𝑅(𝑀𝐵) → 𝐷(𝑋 × 𝑋)(𝑀𝐵)))
7473imp 444 . . . . . . 7 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷(𝑋 × 𝑋)(𝑀𝐵))
75 brxp 5071 . . . . . . . 8 (𝐷(𝑋 × 𝑋)(𝑀𝐵) ↔ (𝐷𝑋 ∧ (𝑀𝐵) ∈ 𝑋))
7675simplbi 475 . . . . . . 7 (𝐷(𝑋 × 𝑋)(𝑀𝐵) → 𝐷𝑋)
7774, 76syl 17 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷𝑋)
7877adantrl 748 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐷𝑋)
79 isorel 6476 . . . . 5 ((𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝑅𝐷 ↔ (𝑀𝐶) E (𝑀𝐷)))
8069, 70, 78, 79syl12anc 1316 . . . 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 6480 . . . . . 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 749 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐶𝑌)
8563simp2d 1067 . . . . . 6 ((𝜑𝐷𝑅(𝑀𝐵)) → 𝐷𝑌)
8685adantrl 748 . . . . 5 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → 𝐷𝑌)
87 isorel 6476 . . . . 5 ((𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝐶𝑌𝐷𝑌)) → (𝐶𝑆𝐷 ↔ (𝑁𝐶) E (𝑁𝐷)))
8883, 84, 86, 87syl12anc 1316 . . . 4 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝐶𝑆𝐷 ↔ (𝑁𝐶) E (𝑁𝐷)))
8966, 80, 883bitr4d 299 . . 3 ((𝜑 ∧ (𝐶𝑅(𝑀𝐵) ∧ 𝐷𝑅(𝑀𝐵))) → (𝐶𝑅𝐷𝐶𝑆𝐷))
9089expr 641 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583  {copab 4642   E cep 4947   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549  OrdIsocoi 8297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-wrecs 7294  df-recs 7355  df-oi 8298 This theorem is referenced by:  fpwwe2lem8  9338
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