Theorem idfu1stf1o 50025

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–1-1-onto→𝐵)

Proof of Theorem idfu1stf1o

Step	Hyp	Ref	Expression
1		f1oi 6856	. 2 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
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 17968	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼) = ( I ↾ 𝐵))
6	5	f1oeq1d 6812	. 2 ⊢ (𝐶 ∈ Cat → ((1st ‘𝐼):𝐵–1-1-onto→𝐵 ↔ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵))
7	1, 6	mpbiri 261	1 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   I cid 5549   ↾ cres 5657  –1-1-onto→wf1o 6532  ‘cfv 6533  1^st c1st 7984  Basecbs 17301  Catccat 17752  id_funccidfu 17944

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-1st 7986  df-idfu 17948

This theorem is used by:  idemb  50085