Theorem homfval 17629

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 17627	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
6		oveq12 7379	. . 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 7405	. 2 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ V)
11	5, 7, 8, 9, 10	ovmpod 7522	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  Basecbs 17150  Hom chom 17202  Hom_f chomf 17603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-homf 17607

This theorem is referenced by:  homfeqval  17634  comfffval2  17638  comffval2  17639  comfval2  17640  catsubcat  17777  subcss2  17781  fullsubc  17788  fullresc  17789  funcres2c  17841  hof1  18191  hofcllem  18195  hofcl  18196  yonffthlem  18219  srhmsubc  20630  srhmsubcALTV  48714  oppcendc  49406  discsubc  49452  ssccatid  49460  imaidfu  49498  imasubc  49539  imassc  49541  setc1onsubc  49990