Theorem idfu1stf1o 49596

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 6812	. 2 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵
2		idfu1stf1o.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐶)
3		idfu1stf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5	2, 3, 4	idfu1st 17844	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼) = ( I ↾ 𝐵))
6	5	f1oeq1d 6769	. 2 ⊢ (𝐶 ∈ Cat → ((1st ‘𝐼):𝐵–1-1-onto→𝐵 ↔ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵))
7	1, 6	mpbiri 259	1 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   I cid 5519   ↾ cres 5627  –1-1-onto→wf1o 6491  ‘cfv 6492  1^st c1st 7936  Basecbs 17177  Catccat 17628  id_funccidfu 17820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7938  df-idfu 17824

This theorem is referenced by:  idemb  49656