Theorem homfval 17660

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

Hypotheses

Ref	Expression
homffval.f	⊢ 𝐹 = (Homf ‘𝐶)
homffval.b	⊢ 𝐵 = (Base‘𝐶)
homffval.h	⊢ 𝐻 = (Hom ‘𝐶)
homfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
homfval	⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Proof of Theorem homfval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homffval.f	. . . 4 ⊢ 𝐹 = (Homf ‘𝐶)
2		homffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		homffval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	homffval 17658	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
6		oveq12 7399	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
7	6	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
8		homfval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		homfval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		ovexd 7425	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V)
11	5, 7, 8, 9, 10	ovmpod 7544	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17186  Hom chom 17238  Hom_f chomf 17634

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-homf 17638

This theorem is referenced by:  homfeqval  17665  comfffval2  17669  comffval2  17670  comfval2  17671  catsubcat  17808  subcss2  17812  fullsubc  17819  fullresc  17820  funcres2c  17872  hof1  18222  hofcllem  18226  hofcl  18227  yonffthlem  18250  srhmsubc  20596  srhmsubcALTV  48317  oppcendc  49011  discsubc  49057  ssccatid  49065  imaidfu  49103  imasubc  49144  imassc  49146  setc1onsubc  49595