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Theorem hof2val 18219
Description: The morphism part of the Hom functor, for morphisms βŸ¨π‘“, π‘”βŸ©:βŸ¨π‘‹, π‘ŒβŸ©βŸΆβŸ¨π‘, π‘ŠβŸ© (which since the first argument is contravariant means morphisms 𝑓:π‘βŸΆπ‘‹ and 𝑔:π‘ŒβŸΆπ‘Š), yields a function (a morphism of SetCat) mapping β„Ž:π‘‹βŸΆπ‘Œ to 𝑔 ∘ β„Ž ∘ 𝑓:π‘βŸΆπ‘Š. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m 𝑀 = (HomFβ€˜πΆ)
hofval.c (πœ‘ β†’ 𝐢 ∈ Cat)
hof1.b 𝐡 = (Baseβ€˜πΆ)
hof1.h 𝐻 = (Hom β€˜πΆ)
hof1.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
hof1.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
hof2.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
hof2.w (πœ‘ β†’ π‘Š ∈ 𝐡)
hof2.o Β· = (compβ€˜πΆ)
hof2.f (πœ‘ β†’ 𝐹 ∈ (𝑍𝐻𝑋))
hof2.g (πœ‘ β†’ 𝐺 ∈ (π‘Œπ»π‘Š))
Assertion
Ref Expression
hof2val (πœ‘ β†’ (𝐹(βŸ¨π‘‹, π‘ŒβŸ©(2nd β€˜π‘€)βŸ¨π‘, π‘ŠβŸ©)𝐺) = (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹)))
Distinct variable groups:   𝐡,β„Ž   β„Ž,𝐹   β„Ž,𝐺   πœ‘,β„Ž   𝐢,β„Ž   β„Ž,𝐻   β„Ž,π‘Š   Β· ,β„Ž   β„Ž,𝑋   β„Ž,π‘Œ   β„Ž,𝑍
Allowed substitution hint:   𝑀(β„Ž)

Proof of Theorem hof2val
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3 𝑀 = (HomFβ€˜πΆ)
2 hofval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
3 hof1.b . . 3 𝐡 = (Baseβ€˜πΆ)
4 hof1.h . . 3 𝐻 = (Hom β€˜πΆ)
5 hof1.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 hof1.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 hof2.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
8 hof2.w . . 3 (πœ‘ β†’ π‘Š ∈ 𝐡)
9 hof2.o . . 3 Β· = (compβ€˜πΆ)
101, 2, 3, 4, 5, 6, 7, 8, 9hof2fval 18218 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ©(2nd β€˜π‘€)βŸ¨π‘, π‘ŠβŸ©) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (π‘Œπ»π‘Š) ↦ (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝑓))))
11 simplrr 775 . . . . 5 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ β„Ž ∈ (π‘‹π»π‘Œ)) β†’ 𝑔 = 𝐺)
1211oveq1d 7419 . . . 4 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ β„Ž ∈ (π‘‹π»π‘Œ)) β†’ (𝑔(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž) = (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž))
13 simplrl 774 . . . 4 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ β„Ž ∈ (π‘‹π»π‘Œ)) β†’ 𝑓 = 𝐹)
1412, 13oveq12d 7422 . . 3 (((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ β„Ž ∈ (π‘‹π»π‘Œ)) β†’ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝑓) = ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹))
1514mpteq2dva 5241 . 2 ((πœ‘ ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) β†’ (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝑔(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝑓)) = (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹)))
16 hof2.f . 2 (πœ‘ β†’ 𝐹 ∈ (𝑍𝐻𝑋))
17 hof2.g . 2 (πœ‘ β†’ 𝐺 ∈ (π‘Œπ»π‘Š))
18 ovex 7437 . . . 4 (π‘‹π»π‘Œ) ∈ V
1918mptex 7219 . . 3 (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹)) ∈ V
2019a1i 11 . 2 (πœ‘ β†’ (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹)) ∈ V)
2110, 15, 16, 17, 20ovmpod 7555 1 (πœ‘ β†’ (𝐹(βŸ¨π‘‹, π‘ŒβŸ©(2nd β€˜π‘€)βŸ¨π‘, π‘ŠβŸ©)𝐺) = (β„Ž ∈ (π‘‹π»π‘Œ) ↦ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· π‘Š)β„Ž)(βŸ¨π‘, π‘‹βŸ© Β· π‘Š)𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βŸ¨cop 4629   ↦ cmpt 5224  β€˜cfv 6536  (class class class)co 7404  2nd c2nd 7970  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  HomFchof 18211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-hof 18213
This theorem is referenced by:  hof2  18220  hofcllem  18221  hofcl  18222  yonedalem3b  18242
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