Theorem idfu1stf1o 50546

Description: The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025.)

Hypotheses

Ref	Expression
idfu1stf1o.i	⊢ 𝐼 = (idfunc‘𝐶)
idfu1stf1o.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
idfu1stf1o	⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵⟶1-1onto𝐵)

Proof of Theorem idfu1stf1o

Step	Hyp	Ref	Expression
1		f1oi 6894	. 2 ⊢ ( I ↾ 𝐵):𝐵⟶1-1onto𝐵
2		idfu1stf1o.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3		idfu1stf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		id 23	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5	2, 3, 4	idfu1st 18104	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼) = ( I ↾ 𝐵))
6	5	f1oeq1d 6849	. 2 ⊢ (𝐶 ∈ Cat → ((1st ‘𝐼):𝐵⟶1-1onto𝐵 ↔ ( I ↾ 𝐵):𝐵⟶1-1onto𝐵))
7	1, 6	mpbiri 261	1 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵⟶1-1onto𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   I cid 5552   ↾ cres 5660  ⟶1-1ontowf1o 6565  ‘cfv 6566  1^st c1st 8033  Basecbs 17431  Catccat 17888  id_funccidfu 18080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1st 8035  df-idfu 18084

This theorem is used by:  idemb  50606