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Theorem comfffval 16274
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o 𝑂 = (compf𝐶)
comfffval.b 𝐵 = (Base‘𝐶)
comfffval.h 𝐻 = (Hom ‘𝐶)
comfffval.x · = (comp‘𝐶)
Assertion
Ref Expression
comfffval 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑓,𝑔,𝑥,𝑦,𝐶   · ,𝑓,𝑔,𝑥   𝑓,𝐻,𝑔,𝑥
Allowed substitution hints:   𝐵(𝑓,𝑔)   · (𝑦)   𝐻(𝑦)   𝑂(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem comfffval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 comfffval.o . 2 𝑂 = (compf𝐶)
2 fveq2 6150 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 comfffval.b . . . . . . 7 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2678 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
54sqxpeqd 5106 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
6 fveq2 6150 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 comfffval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2678 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 6622 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑥)(Hom ‘𝑐)𝑦) = ((2nd𝑥)𝐻𝑦))
108fveq1d 6152 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
11 fveq2 6150 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
12 comfffval.x . . . . . . . . 9 · = (comp‘𝐶)
1311, 12syl6eqr 2678 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1413oveqd 6622 . . . . . . 7 (𝑐 = 𝐶 → (𝑥(comp‘𝑐)𝑦) = (𝑥 · 𝑦))
1514oveqd 6622 . . . . . 6 (𝑐 = 𝐶 → (𝑔(𝑥(comp‘𝑐)𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
169, 10, 15mpt2eq123dv 6671 . . . . 5 (𝑐 = 𝐶 → (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)) = (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
175, 4, 16mpt2eq123dv 6671 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
18 df-comf 16248 . . . 4 compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
19 fvex 6160 . . . . . . 7 (Base‘𝐶) ∈ V
203, 19eqeltri 2700 . . . . . 6 𝐵 ∈ V
2120, 20xpex 6916 . . . . 5 (𝐵 × 𝐵) ∈ V
2221, 20mpt2ex 7193 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) ∈ V
2317, 18, 22fvmpt 6240 . . 3 (𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
24 fvprc 6144 . . . 4 𝐶 ∈ V → (compf𝐶) = ∅)
25 fvprc 6144 . . . . . . . . 9 𝐶 ∈ V → (Base‘𝐶) = ∅)
263, 25syl5eq 2672 . . . . . . . 8 𝐶 ∈ V → 𝐵 = ∅)
2726xpeq2d 5104 . . . . . . 7 𝐶 ∈ V → (𝐵 × 𝐵) = (𝐵 × ∅))
28 xp0 5515 . . . . . . 7 (𝐵 × ∅) = ∅
2927, 28syl6eq 2676 . . . . . 6 𝐶 ∈ V → (𝐵 × 𝐵) = ∅)
30 mpt2eq12 6669 . . . . . 6 (((𝐵 × 𝐵) = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
3129, 26, 30syl2anc 692 . . . . 5 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
32 mpt20 6679 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅
3331, 32syl6eq 2676 . . . 4 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
3424, 33eqtr4d 2663 . . 3 𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
3523, 34pm2.61i 176 . 2 (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
361, 35eqtri 2648 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1992  Vcvv 3191  c0 3896   × cxp 5077  cfv 5850  (class class class)co 6605  cmpt2 6607  2nd c2nd 7115  Basecbs 15776  Hom chom 15868  compcco 15869  compfccomf 16244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-comf 16248
This theorem is referenced by:  comffval  16275  comfffval2  16277  comfffn  16280  comfeq  16282
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