Theorem isofnALT 49957

Description: The function value of the function returning the isomorphisms of a category is a function over the Cartesian square of the base set of the category. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
isofnALT	⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Proof of Theorem isofnALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7898	. . . . . 6 ⊢ (𝑥 ∈ V → dom 𝑥 ∈ V)
2	1	adantl 487	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ V) → dom 𝑥 ∈ V)
3	2	ralrimiva 3154	. . . 4 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ V dom 𝑥 ∈ V)
4		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ V ↦ dom 𝑥) = (𝑥 ∈ V ↦ dom 𝑥)
5	4	fnmpt 6672	. . . 4 ⊢ (∀𝑥 ∈ V dom 𝑥 ∈ V → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
6	3, 5	syl 18	. . 3 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
7		invfn 49956	. . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		ssv 3955	. . . 4 ⊢ ran (Inv‘𝐶) ⊆ V
9	8	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ran (Inv‘𝐶) ⊆ V)
10		fnco 6650	. . 3 ⊢ (((𝑥 ∈ V ↦ dom 𝑥) Fn V ∧ (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Inv‘𝐶) ⊆ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
11	6, 7, 9, 10	syl3anc 1398	. 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
12		isofval 17846	. . 3 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
13	12	fneq1d 6625	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶))))
14	11, 13	mpbird 260	1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899   ↦ cmpt 5186   × cxp 5653  dom cdm 5655  ran crn 5656   ∘ ccom 5659   Fn wfn 6528  ‘cfv 6533  Basecbs 17301  Catccat 17752  Invcinv 17834  Isociso 17835

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-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-inv 17837  df-iso 17838

This theorem is used by: (None)