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Theorem soisores 6542
 Description: Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
Assertion
Ref Expression
soisores (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem soisores
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isorel 6541 . . . . 5 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦)))
2 fvres 6174 . . . . . . 7 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
3 fvres 6174 . . . . . . 7 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
42, 3breqan12d 4639 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
54adantl 482 . . . . 5 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
61, 5bitrd 268 . . . 4 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
76biimpd 219 . . 3 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
87ralrimivva 2967 . 2 ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
9 ffn 6012 . . . . . . . 8 (𝐹:𝐵𝐶𝐹 Fn 𝐵)
109ad2antrl 763 . . . . . . 7 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → 𝐹 Fn 𝐵)
11 simprr 795 . . . . . . 7 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → 𝐴𝐵)
12 fnssres 5972 . . . . . . 7 ((𝐹 Fn 𝐵𝐴𝐵) → (𝐹𝐴) Fn 𝐴)
1310, 11, 12syl2anc 692 . . . . . 6 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → (𝐹𝐴) Fn 𝐴)
14133adant3 1079 . . . . 5 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → (𝐹𝐴) Fn 𝐴)
15 df-ima 5097 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
1615eqcomi 2630 . . . . . 6 ran (𝐹𝐴) = (𝐹𝐴)
1716a1i 11 . . . . 5 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → ran (𝐹𝐴) = (𝐹𝐴))
18 fvres 6174 . . . . . . . . 9 (𝑧𝐴 → ((𝐹𝐴)‘𝑧) = (𝐹𝑧))
19 fvres 6174 . . . . . . . . 9 (𝑤𝐴 → ((𝐹𝐴)‘𝑤) = (𝐹𝑤))
2018, 19eqeqan12d 2637 . . . . . . . 8 ((𝑧𝐴𝑤𝐴) → (((𝐹𝐴)‘𝑧) = ((𝐹𝐴)‘𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
2120adantl 482 . . . . . . 7 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (((𝐹𝐴)‘𝑧) = ((𝐹𝐴)‘𝑤) ↔ (𝐹𝑧) = (𝐹𝑤)))
22 simprl 793 . . . . . . . . . . 11 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑧𝐴)
23 simprr 795 . . . . . . . . . . 11 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑤𝐴)
24 simpl3 1064 . . . . . . . . . . 11 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
25 breq1 4626 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
26 fveq2 6158 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2726breq1d 4633 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
2825, 27imbi12d 334 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)) ↔ (𝑧𝑅𝑦 → (𝐹𝑧)𝑆(𝐹𝑦))))
29 breq2 4627 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑧𝑅𝑦𝑧𝑅𝑤))
30 fveq2 6158 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
3130breq2d 4635 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
3229, 31imbi12d 334 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑧𝑅𝑦 → (𝐹𝑧)𝑆(𝐹𝑦)) ↔ (𝑧𝑅𝑤 → (𝐹𝑧)𝑆(𝐹𝑤))))
3328, 32rspc2va 3312 . . . . . . . . . . 11 (((𝑧𝐴𝑤𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → (𝑧𝑅𝑤 → (𝐹𝑧)𝑆(𝐹𝑤)))
3422, 23, 24, 33syl21anc 1322 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧𝑅𝑤 → (𝐹𝑧)𝑆(𝐹𝑤)))
35 breq1 4626 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥𝑅𝑦𝑤𝑅𝑦))
36 fveq2 6158 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
3736breq1d 4633 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑦)))
3835, 37imbi12d 334 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)) ↔ (𝑤𝑅𝑦 → (𝐹𝑤)𝑆(𝐹𝑦))))
39 breq2 4627 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑤𝑅𝑦𝑤𝑅𝑧))
40 fveq2 6158 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
4140breq2d 4635 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ((𝐹𝑤)𝑆(𝐹𝑦) ↔ (𝐹𝑤)𝑆(𝐹𝑧)))
4239, 41imbi12d 334 . . . . . . . . . . . 12 (𝑦 = 𝑧 → ((𝑤𝑅𝑦 → (𝐹𝑤)𝑆(𝐹𝑦)) ↔ (𝑤𝑅𝑧 → (𝐹𝑤)𝑆(𝐹𝑧))))
4338, 42rspc2va 3312 . . . . . . . . . . 11 (((𝑤𝐴𝑧𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → (𝑤𝑅𝑧 → (𝐹𝑤)𝑆(𝐹𝑧)))
4423, 22, 24, 43syl21anc 1322 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑤𝑅𝑧 → (𝐹𝑤)𝑆(𝐹𝑧)))
4534, 44orim12d 882 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝑧𝑅𝑤𝑤𝑅𝑧) → ((𝐹𝑧)𝑆(𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
4645con3d 148 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (¬ ((𝐹𝑧)𝑆(𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧)) → ¬ (𝑧𝑅𝑤𝑤𝑅𝑧)))
47 simpl1r 1111 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑆 Or 𝐶)
48 simpl2l 1112 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝐹:𝐵𝐶)
49 simpl2r 1113 . . . . . . . . . . 11 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝐴𝐵)
5049, 22sseldd 3589 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑧𝐵)
5148, 50ffvelrnd 6326 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝐹𝑧) ∈ 𝐶)
5249, 23sseldd 3589 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑤𝐵)
5348, 52ffvelrnd 6326 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝐹𝑤) ∈ 𝐶)
54 sotrieq 5032 . . . . . . . . 9 ((𝑆 Or 𝐶 ∧ ((𝐹𝑧) ∈ 𝐶 ∧ (𝐹𝑤) ∈ 𝐶)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ¬ ((𝐹𝑧)𝑆(𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
5547, 51, 53, 54syl12anc 1321 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ¬ ((𝐹𝑧)𝑆(𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
56 simpl1l 1110 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → 𝑅 Or 𝐵)
57 sotrieq 5032 . . . . . . . . 9 ((𝑅 Or 𝐵 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤𝑤𝑅𝑧)))
5856, 50, 52, 57syl12anc 1321 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤𝑤𝑅𝑧)))
5946, 55, 583imtr4d 283 . . . . . . 7 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
6021, 59sylbid 230 . . . . . 6 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (((𝐹𝐴)‘𝑧) = ((𝐹𝐴)‘𝑤) → 𝑧 = 𝑤))
6160ralrimivva 2967 . . . . 5 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → ∀𝑧𝐴𝑤𝐴 (((𝐹𝐴)‘𝑧) = ((𝐹𝐴)‘𝑤) → 𝑧 = 𝑤))
62 dff1o6 6496 . . . . 5 ((𝐹𝐴):𝐴1-1-onto→(𝐹𝐴) ↔ ((𝐹𝐴) Fn 𝐴 ∧ ran (𝐹𝐴) = (𝐹𝐴) ∧ ∀𝑧𝐴𝑤𝐴 (((𝐹𝐴)‘𝑧) = ((𝐹𝐴)‘𝑤) → 𝑧 = 𝑤)))
6314, 17, 61, 62syl3anbrc 1244 . . . 4 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴))
64 fveq2 6158 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
6564a1i 11 . . . . . . . . . 10 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤)))
6665, 44orim12d 882 . . . . . . . . 9 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝑧 = 𝑤𝑤𝑅𝑧) → ((𝐹𝑧) = (𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
6766con3d 148 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (¬ ((𝐹𝑧) = (𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧)) → ¬ (𝑧 = 𝑤𝑤𝑅𝑧)))
68 sotric 5031 . . . . . . . . 9 ((𝑆 Or 𝐶 ∧ ((𝐹𝑧) ∈ 𝐶 ∧ (𝐹𝑤) ∈ 𝐶)) → ((𝐹𝑧)𝑆(𝐹𝑤) ↔ ¬ ((𝐹𝑧) = (𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
6947, 51, 53, 68syl12anc 1321 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝐹𝑧)𝑆(𝐹𝑤) ↔ ¬ ((𝐹𝑧) = (𝐹𝑤) ∨ (𝐹𝑤)𝑆(𝐹𝑧))))
70 sotric 5031 . . . . . . . . 9 ((𝑅 Or 𝐵 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤𝑤𝑅𝑧)))
7156, 50, 52, 70syl12anc 1321 . . . . . . . 8 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤𝑤𝑅𝑧)))
7267, 69, 713imtr4d 283 . . . . . . 7 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → ((𝐹𝑧)𝑆(𝐹𝑤) → 𝑧𝑅𝑤))
7334, 72impbid 202 . . . . . 6 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
7418, 19breqan12d 4639 . . . . . . 7 ((𝑧𝐴𝑤𝐴) → (((𝐹𝐴)‘𝑧)𝑆((𝐹𝐴)‘𝑤) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
7574adantl 482 . . . . . 6 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (((𝐹𝐴)‘𝑧)𝑆((𝐹𝐴)‘𝑤) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
7673, 75bitr4d 271 . . . . 5 ((((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ∧ (𝑧𝐴𝑤𝐴)) → (𝑧𝑅𝑤 ↔ ((𝐹𝐴)‘𝑧)𝑆((𝐹𝐴)‘𝑤)))
7776ralrimivva 2967 . . . 4 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ ((𝐹𝐴)‘𝑧)𝑆((𝐹𝐴)‘𝑤)))
78 df-isom 5866 . . . 4 ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ↔ ((𝐹𝐴):𝐴1-1-onto→(𝐹𝐴) ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ ((𝐹𝐴)‘𝑧)𝑆((𝐹𝐴)‘𝑤))))
7963, 77, 78sylanbrc 697 . . 3 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) → (𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)))
80793expia 1264 . 2 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)) → (𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴))))
818, 80impbid2 216 1 (((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2908   ⊆ wss 3560   class class class wbr 4623   Or wor 5004  ran crn 5085   ↾ cres 5086   “ cima 5087   Fn wfn 5852  ⟶wf 5853  –1-1-onto→wf1o 5856  ‘cfv 5857   Isom wiso 5858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866 This theorem is referenced by:  isercolllem1  14345  dvgt0lem2  23704  erdszelem4  30937  erdszelem8  30941  erdsze2lem2  30947
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