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Theorem comfffval2 16408
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o 𝑂 = (compf𝐶)
comfffval2.b 𝐵 = (Base‘𝐶)
comfffval2.h 𝐻 = (Homf𝐶)
comfffval2.x · = (comp‘𝐶)
Assertion
Ref Expression
comfffval2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐵   𝐶,𝑓,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥
Allowed substitution hints:   · (𝑦)   𝐻(𝑥,𝑦,𝑓,𝑔)   𝑂(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem comfffval2
StepHypRef Expression
1 comfffval2.o . . 3 𝑂 = (compf𝐶)
2 comfffval2.b . . 3 𝐵 = (Base‘𝐶)
3 eqid 2651 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
51, 2, 3, 4comfffval 16405 . 2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
6 comfffval2.h . . . . 5 𝐻 = (Homf𝐶)
7 xp2nd 7243 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → (2nd𝑥) ∈ 𝐵)
87adantr 480 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (2nd𝑥) ∈ 𝐵)
9 simpr 476 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
106, 2, 3, 8, 9homfval 16399 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((2nd𝑥)𝐻𝑦) = ((2nd𝑥)(Hom ‘𝐶)𝑦))
11 xp1st 7242 . . . . . . . 8 (𝑥 ∈ (𝐵 × 𝐵) → (1st𝑥) ∈ 𝐵)
1211adantr 480 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (1st𝑥) ∈ 𝐵)
136, 2, 3, 12, 8homfval 16399 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((1st𝑥)𝐻(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
14 df-ov 6693 . . . . . 6 ((1st𝑥)𝐻(2nd𝑥)) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩)
15 df-ov 6693 . . . . . 6 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
1613, 14, 153eqtr3g 2708 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
17 1st2nd2 7249 . . . . . . 7 (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1817adantr 480 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1918fveq2d 6233 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩))
2018fveq2d 6233 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2116, 19, 203eqtr4d 2695 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = ((Hom ‘𝐶)‘𝑥))
22 eqidd 2652 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
2310, 21, 22mpt2eq123dv 6759 . . 3 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)) = (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
2423mpt2eq3ia 6762 . 2 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
255, 24eqtr4i 2676 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   × cxp 5141  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  Hom chom 15999  compcco 16000  Homf chomf 16374  compfccomf 16375 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-homf 16378  df-comf 16379 This theorem is referenced by: (None)
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