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Theorem comfffval 16565
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 6332 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 comfffval.b . . . . . . 7 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2823 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
54sqxpeqd 5281 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
6 fveq2 6332 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 comfffval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2823 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 6810 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑥)(Hom ‘𝑐)𝑦) = ((2nd𝑥)𝐻𝑦))
108fveq1d 6334 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
11 fveq2 6332 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
12 comfffval.x . . . . . . . . 9 · = (comp‘𝐶)
1311, 12syl6eqr 2823 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1413oveqd 6810 . . . . . . 7 (𝑐 = 𝐶 → (𝑥(comp‘𝑐)𝑦) = (𝑥 · 𝑦))
1514oveqd 6810 . . . . . 6 (𝑐 = 𝐶 → (𝑔(𝑥(comp‘𝑐)𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
169, 10, 15mpt2eq123dv 6864 . . . . 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 6864 . . . 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 16539 . . . 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 6342 . . . . . . 7 (Base‘𝐶) ∈ V
203, 19eqeltri 2846 . . . . . 6 𝐵 ∈ V
2120, 20xpex 7109 . . . . 5 (𝐵 × 𝐵) ∈ V
2221, 20mpt2ex 7397 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) ∈ V
2317, 18, 22fvmpt 6424 . . 3 (𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
24 fvprc 6326 . . . 4 𝐶 ∈ V → (compf𝐶) = ∅)
25 fvprc 6326 . . . . . . . . 9 𝐶 ∈ V → (Base‘𝐶) = ∅)
263, 25syl5eq 2817 . . . . . . . 8 𝐶 ∈ V → 𝐵 = ∅)
2726xpeq2d 5279 . . . . . . 7 𝐶 ∈ V → (𝐵 × 𝐵) = (𝐵 × ∅))
28 xp0 5693 . . . . . . 7 (𝐵 × ∅) = ∅
2927, 28syl6eq 2821 . . . . . 6 𝐶 ∈ V → (𝐵 × 𝐵) = ∅)
30 mpt2eq12 6862 . . . . . 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 573 . . . . 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 6872 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅
3331, 32syl6eq 2821 . . . 4 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
3424, 33eqtr4d 2808 . . 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 2793 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 1631  wcel 2145  Vcvv 3351  c0 4063   × cxp 5247  cfv 6031  (class class class)co 6793  cmpt2 6795  2nd c2nd 7314  Basecbs 16064  Hom chom 16160  compcco 16161  compfccomf 16535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-comf 16539
This theorem is referenced by:  comffval  16566  comfffval2  16568  comfffn  16571  comfeq  16573
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