Theorem homfval 17737

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 17735	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
6		oveq12 7440	. . 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 7466	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V)
11	5, 7, 8, 9, 10	ovmpod 7585	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  Hom chom 17309  Hom_f chomf 17711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-homf 17715

This theorem is referenced by:  homfeqval  17742  comfffval2  17746  comffval2  17747  comfval2  17748  catsubcat  17890  subcss2  17894  fullsubc  17901  fullresc  17902  funcres2c  17955  hof1  18311  hofcllem  18315  hofcl  18316  yonffthlem  18339  srhmsubc  20697  srhmsubcALTV  48169