Theorem homfeqval 16278

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
homfeqval.b	⊢ 𝐵 = (Base‘𝐶)
homfeqval.h	⊢ 𝐻 = (Hom ‘𝐶)
homfeqval.j	⊢ 𝐽 = (Hom ‘𝐷)
homfeqval.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
homfeqval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homfeqval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
homfeqval	⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))

Proof of Theorem homfeqval

Step	Hyp	Ref	Expression
1		homfeqval.1	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	oveqd 6621	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋(Homf ‘𝐷)𝑌))
3		eqid 2621	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		homfeqval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		homfeqval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
6		homfeqval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		homfeqval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	3, 4, 5, 6, 7	homfval 16273	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
9		eqid 2621	. . 3 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
10		eqid 2621	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
11		homfeqval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
12	1	homfeqbas 16277	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
13	4, 12	syl5eq 2667	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
14	6, 13	eleqtrd 2700	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
15	7, 13	eleqtrd 2700	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
16	9, 10, 11, 14, 15	homfval 16273	. 2 ⊢ (𝜑 → (𝑋(Homf ‘𝐷)𝑌) = (𝑋𝐽𝑌))
17	2, 8, 16	3eqtr3d 2663	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))

Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  Hom_f chomf 16248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-homf 16252

This theorem is referenced by:  comfeq  16287  comfeqval  16289  catpropd  16290  cidpropd  16291  monpropd  16318  funcpropd  16481  fullpropd  16501  natpropd  16557  xpcpropd  16769  curfpropd  16794  hofpropd  16828