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Theorem comfffval 16943
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
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 6642 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 comfffval.b . . . . . . 7 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2873 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
54sqxpeqd 5559 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
6 fveq2 6642 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 comfffval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2873 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 7146 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑥)(Hom ‘𝑐)𝑦) = ((2nd𝑥)𝐻𝑦))
108fveq1d 6644 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
11 fveq2 6642 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
12 comfffval.x . . . . . . . . 9 · = (comp‘𝐶)
1311, 12syl6eqr 2873 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1413oveqd 7146 . . . . . . 7 (𝑐 = 𝐶 → (𝑥(comp‘𝑐)𝑦) = (𝑥 · 𝑦))
1514oveqd 7146 . . . . . 6 (𝑐 = 𝐶 → (𝑔(𝑥(comp‘𝑐)𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
169, 10, 15mpoeq123dv 7202 . . . . 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, 16mpoeq123dv 7202 . . . 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 16917 . . . 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))))
193fvexi 6656 . . . . . 6 𝐵 ∈ V
2019, 19xpex 7450 . . . . 5 (𝐵 × 𝐵) ∈ V
2120, 19mpoex 7751 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) ∈ V
2217, 18, 21fvmpt 6740 . . 3 (𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
23 fvprc 6635 . . . 4 𝐶 ∈ V → (compf𝐶) = ∅)
24 fvprc 6635 . . . . . . 7 𝐶 ∈ V → (Base‘𝐶) = ∅)
253, 24syl5eq 2867 . . . . . 6 𝐶 ∈ V → 𝐵 = ∅)
2625olcd 870 . . . . 5 𝐶 ∈ V → ((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅))
27 0mpo0 7210 . . . . 5 (((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
2826, 27syl 17 . . . 4 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
2923, 28eqtr4d 2858 . . 3 𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
3022, 29pm2.61i 184 . 2 (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
311, 30eqtri 2843 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  wo 843   = wceq 1537  wcel 2114  Vcvv 3470  c0 4265   × cxp 5525  cfv 6327  (class class class)co 7129  cmpo 7131  2nd c2nd 7662  Basecbs 16458  Hom chom 16551  compcco 16552  compfccomf 16913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7132  df-oprab 7133  df-mpo 7134  df-1st 7663  df-2nd 7664  df-comf 16917
This theorem is referenced by:  comffval  16944  comfffval2  16946  comfffn  16949  comfeq  16951
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