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Theorem isof1oidb 7057
 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 6590 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1fveq 6999 . . . . . 6 ((𝐻:𝐴1-1𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
31, 2sylan 583 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
4 fvex 6659 . . . . . . 7 (𝐻𝑦) ∈ V
54ideq 5688 . . . . . 6 ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦))
65a1i 11 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦)))
7 ideqg 5687 . . . . . 6 (𝑦𝐴 → (𝑥 I 𝑦𝑥 = 𝑦))
87ad2antll 728 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦𝑥 = 𝑦))
93, 6, 83bitr4rd 315 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
109ralrimivva 3156 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
1110pm4.71i 563 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
12 df-isom 6334 . 2 (𝐻 Isom I , I (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
1311, 12bitr4i 281 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5031   I cid 5425  –1-1→wf1 6322  –1-1-onto→wf1o 6324  ‘cfv 6325   Isom wiso 6326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-f1o 6332  df-fv 6333  df-isom 6334 This theorem is referenced by: (None)
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