Theorem idfu1stf1o 50309

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 6864	. 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 18060	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼) = ( I ↾ 𝐵))
6	5	f1oeq1d 6820	. 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 1569   ∈ wcel 2144   I cid 5544   ↾ cres 5652  –1-1-onto→wf1o 6538  ‘cfv 6539  1^st c1st 8000  Basecbs 17393  Catccat 17844  id_funccidfu 18036

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-1st 8002  df-idfu 18040

This theorem is used by:  idemb  50369