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Theorem isocnv2 6536
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 ralcom 3095 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
2 vex 3194 . . . . . . 7 𝑥 ∈ V
3 vex 3194 . . . . . . 7 𝑦 ∈ V
42, 3brcnv 5270 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
5 fvex 6160 . . . . . . 7 (𝐻𝑥) ∈ V
6 fvex 6160 . . . . . . 7 (𝐻𝑦) ∈ V
75, 6brcnv 5270 . . . . . 6 ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑦)𝑆(𝐻𝑥))
84, 7bibi12i 329 . . . . 5 ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
982ralbii 2980 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
101, 9bitr4i 267 . . 3 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1110anbi2i 729 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
12 df-isom 5859 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
13 df-isom 5859 . 2 (𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1411, 12, 133bitr4i 292 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wral 2912   class class class wbr 4618  ccnv 5078  1-1-ontowf1o 5849  cfv 5850   Isom wiso 5851
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-cnv 5087  df-iota 5813  df-fv 5858  df-isom 5859
This theorem is referenced by:  infiso  8358  wofib  8395  leiso  13178  gtiso  29312
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