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Theorem isocnv2 6621
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 3127 . . . 4 (∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
2 vex 3234 . . . . . . 7 𝑥 ∈ V
3 vex 3234 . . . . . . 7 𝑦 ∈ V
42, 3brcnv 5337 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
5 fvex 6239 . . . . . . 7 (𝐻𝑥) ∈ V
6 fvex 6239 . . . . . . 7 (𝐻𝑦) ∈ V
75, 6brcnv 5337 . . . . . 6 ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑦)𝑆(𝐻𝑥))
84, 7bibi12i 328 . . . . 5 ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥)))
982ralbii 3010 . . . 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 730 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
12 df-isom 5935 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑥𝐴 (𝑦𝑅𝑥 ↔ (𝐻𝑦)𝑆(𝐻𝑥))))
13 df-isom 5935 . 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 383  wral 2941   class class class wbr 4685  ccnv 5142  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-cnv 5151  df-iota 5889  df-fv 5934  df-isom 5935
This theorem is referenced by:  infiso  8454  wofib  8491  leiso  13281  gtiso  29606
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