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Theorem hof2fval 17281
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 17278 . . 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 6459 . . . 4 (Homf𝐶) ∈ V
83fvexi 6460 . . . . . 6 𝐵 ∈ V
98, 8xpex 7240 . . . . 5 (𝐵 × 𝐵) ∈ V
109, 9mpt2ex 7527 . . . 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 7456 . . 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 763 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑦 = ⟨𝑍, 𝑊⟩)
1413fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = (1st ‘⟨𝑍, 𝑊⟩))
15 hof2.z . . . . . . 7 (𝜑𝑍𝐵)
16 hof2.w . . . . . . 7 (𝜑𝑊𝐵)
17 op1stg 7457 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1815, 16, 17syl2anc 579 . . . . . 6 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1918adantr 474 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
2014, 19eqtrd 2814 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = 𝑍)
21 simprl 761 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2221fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
23 hof1.x . . . . . . 7 (𝜑𝑋𝐵)
24 hof1.y . . . . . . 7 (𝜑𝑌𝐵)
25 op1stg 7457 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2623, 24, 25syl2anc 579 . . . . . 6 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2726adantr 474 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2822, 27eqtrd 2814 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = 𝑋)
2920, 28oveq12d 6940 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((1st𝑦)𝐻(1st𝑥)) = (𝑍𝐻𝑋))
3021fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
31 op2ndg 7458 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3223, 24, 31syl2anc 579 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3332adantr 474 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3430, 33eqtrd 2814 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = 𝑌)
3513fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = (2nd ‘⟨𝑍, 𝑊⟩))
36 op2ndg 7458 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3715, 16, 36syl2anc 579 . . . . . 6 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3837adantr 474 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3935, 38eqtrd 2814 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = 𝑊)
4034, 39oveq12d 6940 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑌𝐻𝑊))
4121fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
42 df-ov 6925 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
4341, 42syl6eqr 2832 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝑋𝐻𝑌))
4420, 28opeq12d 4644 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ⟨(1st𝑦), (1st𝑥)⟩ = ⟨𝑍, 𝑋⟩)
4544, 39oveq12d 6940 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)) = (⟨𝑍, 𝑋· 𝑊))
4621, 39oveq12d 6940 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑥 · (2nd𝑦)) = (⟨𝑋, 𝑌· 𝑊))
4746oveqd 6939 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑔(𝑥 · (2nd𝑦))) = (𝑔(⟨𝑋, 𝑌· 𝑊)))
48 eqidd 2779 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑓 = 𝑓)
4945, 47, 48oveq123d 6943 . . . 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 4969 . . 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, 50mpt2eq123dv 6994 . 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 5393 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
5315, 16opelxpd 5393 . 2 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐵 × 𝐵))
54 ovex 6954 . . . 4 (𝑍𝐻𝑋) ∈ V
55 ovex 6954 . . . 4 (𝑌𝐻𝑊) ∈ V
5654, 55mpt2ex 7527 . . 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, 57ovmpt2d 7065 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 386   = wceq 1601  wcel 2107  Vcvv 3398  cop 4404  cmpt 4965   × cxp 5353  cfv 6135  (class class class)co 6922  cmpt2 6924  1st c1st 7443  2nd c2nd 7444  Basecbs 16255  Hom chom 16349  compcco 16350  Catccat 16710  Homf chomf 16712  HomFchof 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-hof 17276
This theorem is referenced by:  hof2val  17282
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