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Theorem comfffval2 16342
 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 2620 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
51, 2, 3, 4comfffval 16339 . 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 7184 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → (2nd𝑥) ∈ 𝐵)
87adantr 481 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (2nd𝑥) ∈ 𝐵)
9 simpr 477 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
106, 2, 3, 8, 9homfval 16333 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((2nd𝑥)𝐻𝑦) = ((2nd𝑥)(Hom ‘𝐶)𝑦))
11 xp1st 7183 . . . . . . . 8 (𝑥 ∈ (𝐵 × 𝐵) → (1st𝑥) ∈ 𝐵)
1211adantr 481 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (1st𝑥) ∈ 𝐵)
136, 2, 3, 12, 8homfval 16333 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((1st𝑥)𝐻(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
14 df-ov 6638 . . . . . 6 ((1st𝑥)𝐻(2nd𝑥)) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩)
15 df-ov 6638 . . . . . 6 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
1613, 14, 153eqtr3g 2677 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
17 1st2nd2 7190 . . . . . . 7 (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1817adantr 481 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1918fveq2d 6182 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩))
2018fveq2d 6182 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2116, 19, 203eqtr4d 2664 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = ((Hom ‘𝐶)‘𝑥))
22 eqidd 2621 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
2310, 21, 22mpt2eq123dv 6702 . . 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 6705 . 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 2645 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ⟨cop 4174   × cxp 5102  ‘cfv 5876  (class class class)co 6635   ↦ cmpt2 6637  1st c1st 7151  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  compcco 15934  Homf chomf 16308  compfccomf 16309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-homf 16312  df-comf 16313 This theorem is referenced by: (None)
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