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Theorem isof1oidb 6770
Description: A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)
Assertion
Ref Expression
isof1oidb (𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))

Proof of Theorem isof1oidb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of1 6323 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1fveq 6715 . . . . . 6 ((𝐻:𝐴1-1𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
31, 2sylan 575 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
4 fvex 6392 . . . . . . 7 (𝐻𝑦) ∈ V
54ideq 5445 . . . . . 6 ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦))
65a1i 11 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦)))
7 ideqg 5444 . . . . . 6 (𝑦𝐴 → (𝑥 I 𝑦𝑥 = 𝑦))
87ad2antll 720 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦𝑥 = 𝑦))
93, 6, 83bitr4rd 303 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
109ralrimivva 3118 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
1110pm4.71i 555 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
12 df-isom 6079 . 2 (𝐻 Isom I , I (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
1311, 12bitr4i 269 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055   class class class wbr 4811   I cid 5186  1-1wf1 6067  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-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-f1o 6077  df-fv 6078  df-isom 6079
This theorem is referenced by: (None)
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