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Theorem comfffval2 17672
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 2727 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
51, 2, 3, 4comfffval 17669 . 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 8020 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → (2nd𝑥) ∈ 𝐵)
87adantr 480 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (2nd𝑥) ∈ 𝐵)
9 simpr 484 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
106, 2, 3, 8, 9homfval 17663 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((2nd𝑥)𝐻𝑦) = ((2nd𝑥)(Hom ‘𝐶)𝑦))
11 xp1st 8019 . . . . . . . 8 (𝑥 ∈ (𝐵 × 𝐵) → (1st𝑥) ∈ 𝐵)
1211adantr 480 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (1st𝑥) ∈ 𝐵)
136, 2, 3, 12, 8homfval 17663 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((1st𝑥)𝐻(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
14 df-ov 7417 . . . . . 6 ((1st𝑥)𝐻(2nd𝑥)) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩)
15 df-ov 7417 . . . . . 6 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
1613, 14, 153eqtr3g 2790 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
17 1st2nd2 8026 . . . . . . 7 (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1817adantr 480 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1918fveq2d 6895 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩))
2018fveq2d 6895 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2116, 19, 203eqtr4d 2777 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = ((Hom ‘𝐶)‘𝑥))
22 eqidd 2728 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
2310, 21, 22mpoeq123dv 7489 . . 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)))
2423mpoeq3ia 7492 . 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 2758 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  cop 4630   × cxp 5670  cfv 6542  (class class class)co 7414  cmpo 7416  1st c1st 7985  2nd c2nd 7986  Basecbs 17171  Hom chom 17235  compcco 17236  Homf chomf 17637  compfccomf 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-homf 17641  df-comf 17642
This theorem is referenced by: (None)
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