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Theorem isof1oidb 6737
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 6297 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1fveq 6682 . . . . . 6 ((𝐻:𝐴1-1𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
31, 2sylan 489 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
4 fvex 6362 . . . . . . 7 (𝐻𝑦) ∈ V
54ideq 5430 . . . . . 6 ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦))
65a1i 11 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦)))
7 ideqg 5429 . . . . . 6 (𝑦𝐴 → (𝑥 I 𝑦𝑥 = 𝑦))
87ad2antll 767 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦𝑥 = 𝑦))
93, 6, 83bitr4rd 301 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
109ralrimivva 3109 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
1110pm4.71i 667 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
12 df-isom 6058 . 2 (𝐻 Isom I , I (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
1311, 12bitr4i 267 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804   I cid 5173  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-f1o 6056  df-fv 6057  df-isom 6058
This theorem is referenced by: (None)
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