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Theorem isof1oidb 7066
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 6607 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1fveq 7011 . . . . . 6 ((𝐻:𝐴1-1𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
31, 2sylan 580 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
4 fvex 6676 . . . . . . 7 (𝐻𝑦) ∈ V
54ideq 5716 . . . . . 6 ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦))
65a1i 11 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦)))
7 ideqg 5715 . . . . . 6 (𝑦𝐴 → (𝑥 I 𝑦𝑥 = 𝑦))
87ad2antll 725 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦𝑥 = 𝑦))
93, 6, 83bitr4rd 313 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
109ralrimivva 3188 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
1110pm4.71i 560 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
12 df-isom 6357 . 2 (𝐻 Isom I , I (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
1311, 12bitr4i 279 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057   I cid 5452  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-f1o 6355  df-fv 6356  df-isom 6357
This theorem is referenced by: (None)
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