Theorem homfval 17616

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 17614	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
6		oveq12 7362	. . 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 7388	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V)
11	5, 7, 8, 9, 10	ovmpod 7505	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17138  Hom chom 17190  Hom_f chomf 17590

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-homf 17594

This theorem is referenced by:  homfeqval  17621  comfffval2  17625  comffval2  17626  comfval2  17627  catsubcat  17764  subcss2  17768  fullsubc  17775  fullresc  17776  funcres2c  17828  hof1  18178  hofcllem  18182  hofcl  18183  yonffthlem  18206  srhmsubc  20583  srhmsubcALTV  48310  oppcendc  49004  discsubc  49050  ssccatid  49058  imaidfu  49096  imasubc  49137  imassc  49139  setc1onsubc  49588