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Theorem hof2fval 18144
Description: The morphism part of the Hom functor, for morphisms 𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊 (which since the first argument is contravariant means morphisms 𝑓:𝑍𝑋 and 𝑔:𝑌𝑊), yields a function (a morphism of SetCat) mapping :𝑋𝑌 to 𝑔𝑓:𝑍𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m 𝑀 = (HomF𝐶)
hofval.c (𝜑𝐶 ∈ Cat)
hof1.b 𝐵 = (Base‘𝐶)
hof1.h 𝐻 = (Hom ‘𝐶)
hof1.x (𝜑𝑋𝐵)
hof1.y (𝜑𝑌𝐵)
hof2.z (𝜑𝑍𝐵)
hof2.w (𝜑𝑊𝐵)
hof2.o · = (comp‘𝐶)
Assertion
Ref Expression
hof2fval (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
Distinct variable groups:   𝑓,𝑔,,𝐵   𝜑,𝑓,𝑔,   𝐶,𝑓,𝑔,   𝑓,𝐻,𝑔,   𝑓,𝑊,𝑔,   · ,𝑓,𝑔,   𝑓,𝑋,𝑔,   𝑓,𝑌,𝑔,   𝑓,𝑍,𝑔,
Allowed substitution hints:   𝑀(𝑓,𝑔,)

Proof of Theorem hof2fval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . . 4 𝑀 = (HomF𝐶)
2 hofval.c . . . 4 (𝜑𝐶 ∈ Cat)
3 hof1.b . . . 4 𝐵 = (Base‘𝐶)
4 hof1.h . . . 4 𝐻 = (Hom ‘𝐶)
5 hof2.o . . . 4 · = (comp‘𝐶)
61, 2, 3, 4, 5hofval 18141 . . 3 (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)
7 fvex 6855 . . . 4 (Homf𝐶) ∈ V
83fvexi 6856 . . . . . 6 𝐵 ∈ V
98, 8xpex 7687 . . . . 5 (𝐵 × 𝐵) ∈ V
109, 9mpoex 8012 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))) ∈ V
117, 10op2ndd 7932 . . 3 (𝑀 = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩ → (2nd𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))))
126, 11syl 17 . 2 (𝜑 → (2nd𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))))
13 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑦 = ⟨𝑍, 𝑊⟩)
1413fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = (1st ‘⟨𝑍, 𝑊⟩))
15 hof2.z . . . . . . 7 (𝜑𝑍𝐵)
16 hof2.w . . . . . . 7 (𝜑𝑊𝐵)
17 op1stg 7933 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1815, 16, 17syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1918adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
2014, 19eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = 𝑍)
21 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2221fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
23 hof1.x . . . . . . 7 (𝜑𝑋𝐵)
24 hof1.y . . . . . . 7 (𝜑𝑌𝐵)
25 op1stg 7933 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2623, 24, 25syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2726adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2822, 27eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = 𝑋)
2920, 28oveq12d 7375 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((1st𝑦)𝐻(1st𝑥)) = (𝑍𝐻𝑋))
3021fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
31 op2ndg 7934 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3223, 24, 31syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3332adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3430, 33eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = 𝑌)
3513fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = (2nd ‘⟨𝑍, 𝑊⟩))
36 op2ndg 7934 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3715, 16, 36syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3837adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3935, 38eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = 𝑊)
4034, 39oveq12d 7375 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑌𝐻𝑊))
4121fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
42 df-ov 7360 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
4341, 42eqtr4di 2794 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝑋𝐻𝑌))
4420, 28opeq12d 4838 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ⟨(1st𝑦), (1st𝑥)⟩ = ⟨𝑍, 𝑋⟩)
4544, 39oveq12d 7375 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)) = (⟨𝑍, 𝑋· 𝑊))
4621, 39oveq12d 7375 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑥 · (2nd𝑦)) = (⟨𝑋, 𝑌· 𝑊))
4746oveqd 7374 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑔(𝑥 · (2nd𝑦))) = (𝑔(⟨𝑋, 𝑌· 𝑊)))
48 eqidd 2737 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑓 = 𝑓)
4945, 47, 48oveq123d 7378 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓) = ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))
5043, 49mpteq12dv 5196 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓)))
5129, 40, 50mpoeq123dv 7432 . 2 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
5223, 24opelxpd 5671 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
5315, 16opelxpd 5671 . 2 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐵 × 𝐵))
54 ovex 7390 . . . 4 (𝑍𝐻𝑋) ∈ V
55 ovex 7390 . . . 4 (𝑌𝐻𝑊) ∈ V
5654, 55mpoex 8012 . . 3 (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))) ∈ V
5756a1i 11 . 2 (𝜑 → (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))) ∈ V)
5812, 51, 52, 53, 57ovmpod 7507 1 (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cop 4592  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Homf chomf 17546  HomFchof 18137
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-hof 18139
This theorem is referenced by:  hof2val  18145
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