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Theorem hof2fval 17501
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 17498 . . 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 6662 . . . 4 (Homf𝐶) ∈ V
83fvexi 6663 . . . . . 6 𝐵 ∈ V
98, 8xpex 7460 . . . . 5 (𝐵 × 𝐵) ∈ V
109, 9mpoex 7764 . . . 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 7686 . . 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 772 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑦 = ⟨𝑍, 𝑊⟩)
1413fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = (1st ‘⟨𝑍, 𝑊⟩))
15 hof2.z . . . . . . 7 (𝜑𝑍𝐵)
16 hof2.w . . . . . . 7 (𝜑𝑊𝐵)
17 op1stg 7687 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1815, 16, 17syl2anc 587 . . . . . 6 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
1918adantr 484 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
2014, 19eqtrd 2836 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑦) = 𝑍)
21 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2221fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
23 hof1.x . . . . . . 7 (𝜑𝑋𝐵)
24 hof1.y . . . . . . 7 (𝜑𝑌𝐵)
25 op1stg 7687 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2623, 24, 25syl2anc 587 . . . . . 6 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2726adantr 484 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2822, 27eqtrd 2836 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st𝑥) = 𝑋)
2920, 28oveq12d 7157 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((1st𝑦)𝐻(1st𝑥)) = (𝑍𝐻𝑋))
3021fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
31 op2ndg 7688 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3223, 24, 31syl2anc 587 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3332adantr 484 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
3430, 33eqtrd 2836 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑥) = 𝑌)
3513fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = (2nd ‘⟨𝑍, 𝑊⟩))
36 op2ndg 7688 . . . . . . 7 ((𝑍𝐵𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3715, 16, 36syl2anc 587 . . . . . 6 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3837adantr 484 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
3935, 38eqtrd 2836 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd𝑦) = 𝑊)
4034, 39oveq12d 7157 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((2nd𝑥)𝐻(2nd𝑦)) = (𝑌𝐻𝑊))
4121fveq2d 6653 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
42 df-ov 7142 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
4341, 42eqtr4di 2854 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻𝑥) = (𝑋𝐻𝑌))
4420, 28opeq12d 4776 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ⟨(1st𝑦), (1st𝑥)⟩ = ⟨𝑍, 𝑋⟩)
4544, 39oveq12d 7157 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)) = (⟨𝑍, 𝑋· 𝑊))
4621, 39oveq12d 7157 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑥 · (2nd𝑦)) = (⟨𝑋, 𝑌· 𝑊))
4746oveqd 7156 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑔(𝑥 · (2nd𝑦))) = (𝑔(⟨𝑋, 𝑌· 𝑊)))
48 eqidd 2802 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑓 = 𝑓)
4945, 47, 48oveq123d 7160 . . . 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 5118 . . 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 7212 . 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 5561 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
5315, 16opelxpd 5561 . 2 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐵 × 𝐵))
54 ovex 7172 . . . 4 (𝑍𝐻𝑋) ∈ V
55 ovex 7172 . . . 4 (𝑌𝐻𝑊) ∈ V
5654, 55mpoex 7764 . . 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 7285 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 399   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534  cmpt 5113   × cxp 5521  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Homf chomf 16933  HomFchof 17494
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-hof 17496
This theorem is referenced by:  hof2val  17502
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