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Theorem comfffval2 16746
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 2778 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
51, 2, 3, 4comfffval 16743 . 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 7478 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → (2nd𝑥) ∈ 𝐵)
87adantr 474 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (2nd𝑥) ∈ 𝐵)
9 simpr 479 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
106, 2, 3, 8, 9homfval 16737 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((2nd𝑥)𝐻𝑦) = ((2nd𝑥)(Hom ‘𝐶)𝑦))
11 xp1st 7477 . . . . . . . 8 (𝑥 ∈ (𝐵 × 𝐵) → (1st𝑥) ∈ 𝐵)
1211adantr 474 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (1st𝑥) ∈ 𝐵)
136, 2, 3, 12, 8homfval 16737 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((1st𝑥)𝐻(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
14 df-ov 6925 . . . . . 6 ((1st𝑥)𝐻(2nd𝑥)) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩)
15 df-ov 6925 . . . . . 6 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
1613, 14, 153eqtr3g 2837 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
17 1st2nd2 7484 . . . . . . 7 (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1817adantr 474 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1918fveq2d 6450 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩))
2018fveq2d 6450 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2116, 19, 203eqtr4d 2824 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = ((Hom ‘𝐶)‘𝑥))
22 eqidd 2779 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
2310, 21, 22mpt2eq123dv 6994 . . 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 6997 . 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 2805 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wcel 2107  cop 4404   × cxp 5353  cfv 6135  (class class class)co 6922  cmpt2 6924  1st c1st 7443  2nd c2nd 7444  Basecbs 16255  Hom chom 16349  compcco 16350  Homf chomf 16712  compfccomf 16713
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-homf 16716  df-comf 16717
This theorem is referenced by: (None)
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