Theorem idfu1stf1o 49088

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 6838	. 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 17841	. . 3 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼) = ( I ↾ 𝐵))
6	5	f1oeq1d 6795	. 2 ⊢ (𝐶 ∈ Cat → ((1st ‘𝐼):𝐵–1-1-onto→𝐵 ↔ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵))
7	1, 6	mpbiri 258	1 ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   I cid 5532   ↾ cres 5640  –1-1-onto→wf1o 6510  ‘cfv 6511  1^st c1st 7966  Basecbs 17179  Catccat 17625  id_funccidfu 17817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-1st 7968  df-idfu 17821

This theorem is referenced by:  idemb  49148