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Theorem isocnv2 6777
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 3245 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
2 vex 3353 . . . . . . 7 𝑥 ∈ V
3 vex 3353 . . . . . . 7 𝑦 ∈ V
42, 3brcnv 5475 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
5 fvex 6392 . . . . . . 7 (𝐻𝑥) ∈ V
6 fvex 6392 . . . . . . 7 (𝐻𝑦) ∈ V
75, 6brcnv 5475 . . . . . 6 ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑦)𝑆(𝐻𝑥))
84, 7bibi12i 330 . . . . 5 ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
982ralbii 3128 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
101, 9bitr4i 269 . . 3 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
1110anbi2i 616 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
12 df-isom 6079 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
13 df-isom 6079 . 2 (𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
1411, 12, 133bitr4i 294 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wral 3055   class class class wbr 4811  ccnv 5278  1-1-ontowf1o 6069  cfv 6070   Isom wiso 6071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-cnv 5287  df-iota 6033  df-fv 6078  df-isom 6079
This theorem is referenced by:  infiso  8624  wofib  8661  leiso  13451  gtiso  29950
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