MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isof1oidb Structured version   Visualization version   GIF version

Theorem isof1oidb 6528
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 6093 . . . . . 6 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1𝐵)
2 f1fveq 6473 . . . . . 6 ((𝐻:𝐴1-1𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
31, 2sylan 488 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) = (𝐻𝑦) ↔ 𝑥 = 𝑦))
4 fvex 6158 . . . . . . 7 (𝐻𝑦) ∈ V
54ideq 5234 . . . . . 6 ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦))
65a1i 11 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) I (𝐻𝑦) ↔ (𝐻𝑥) = (𝐻𝑦)))
7 ideqg 5233 . . . . . 6 (𝑦𝐴 → (𝑥 I 𝑦𝑥 = 𝑦))
87ad2antll 764 . . . . 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 2965 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦)))
1110pm4.71i 663 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 I 𝑦 ↔ (𝐻𝑥) I (𝐻𝑦))))
12 df-isom 5856 . 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 384   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613   I cid 4984  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-f1o 5854  df-fv 5855  df-isom 5856
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator