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Theorem isocnv2 5681
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
isocnv2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))

Proof of Theorem isocnv2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5676 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
2 f1ofn 5336 . . 3 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
31, 2syl 14 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
4 isof1o 5676 . . 3 (𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
54, 2syl 14 . 2 (𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵) → 𝐻 Fn 𝐴)
6 vex 2663 . . . . . . . . . 10 𝑥 ∈ V
7 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
86, 7brcnv 4692 . . . . . . . . 9 (𝑥𝑅𝑦𝑦𝑅𝑥)
98a1i 9 . . . . . . . 8 (((𝐻 Fn 𝐴𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦𝑦𝑅𝑥))
10 funfvex 5406 . . . . . . . . . . 11 ((Fun 𝐻𝑥 ∈ dom 𝐻) → (𝐻𝑥) ∈ V)
1110funfni 5193 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑥𝐴) → (𝐻𝑥) ∈ V)
1211adantr 274 . . . . . . . . 9 (((𝐻 Fn 𝐴𝑥𝐴) ∧ 𝑦𝐴) → (𝐻𝑥) ∈ V)
13 funfvex 5406 . . . . . . . . . . 11 ((Fun 𝐻𝑦 ∈ dom 𝐻) → (𝐻𝑦) ∈ V)
1413funfni 5193 . . . . . . . . . 10 ((𝐻 Fn 𝐴𝑦𝐴) → (𝐻𝑦) ∈ V)
1514adantlr 468 . . . . . . . . 9 (((𝐻 Fn 𝐴𝑥𝐴) ∧ 𝑦𝐴) → (𝐻𝑦) ∈ V)
16 brcnvg 4690 . . . . . . . . 9 (((𝐻𝑥) ∈ V ∧ (𝐻𝑦) ∈ V) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
1712, 15, 16syl2anc 408 . . . . . . . 8 (((𝐻 Fn 𝐴𝑥𝐴) ∧ 𝑦𝐴) → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
189, 17bibi12d 234 . . . . . . 7 (((𝐻 Fn 𝐴𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
1918ralbidva 2410 . . . . . 6 ((𝐻 Fn 𝐴𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
2019ralbidva 2410 . . . . 5 (𝐻 Fn 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
21 ralcom 2571 . . . . 5 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
2220, 21syl6rbbr 198 . . . 4 (𝐻 Fn 𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2322anbi2d 459 . . 3 (𝐻 Fn 𝐴 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
24 df-isom 5102 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
25 df-isom 5102 . . 3 (𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
2623, 24, 253bitr4g 222 . 2 (𝐻 Fn 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵)))
273, 5, 26pm5.21nii 678 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1465  wral 2393  Vcvv 2660   class class class wbr 3899  ccnv 4508   Fn wfn 5088  1-1-ontowf1o 5092  cfv 5093   Isom wiso 5094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-f1o 5100  df-fv 5101  df-isom 5102
This theorem is referenced by:  infisoti  6887
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