Theorem comfffval 16181

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffval.o	⊢ 𝑂 = (compf‘𝐶)
comfffval.b	⊢ 𝐵 = (Base‘𝐶)
comfffval.h	⊢ 𝐻 = (Hom ‘𝐶)
comfffval.x	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
comfffval	⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (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.

Step	Hyp	Ref	Expression
1		comfffval.o	. 2 ⊢ 𝑂 = (compf‘𝐶)
2		fveq2 6103	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		comfffval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	syl6eqr 2662	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5	4	sqxpeqd 5065	. . . . 5 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
6		fveq2 6103	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7		comfffval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	syl6eqr 2662	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
9	8	oveqd 6566	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((2nd ‘𝑥)(Hom ‘𝑐)𝑦) = ((2nd ‘𝑥)𝐻𝑦))
10	8	fveq1d 6105	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐻‘𝑥))
11		fveq2 6103	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
12		comfffval.x	. . . . . . . . 9 ⊢ · = (comp‘𝐶)
13	11, 12	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
14	13	oveqd 6566	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑥(comp‘𝑐)𝑦) = (𝑥 · 𝑦))
15	14	oveqd 6566	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑔(𝑥(comp‘𝑐)𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
16	9, 10, 15	mpt2eq123dv 6615	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)) = (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
17	5, 4, 16	mpt2eq123dv 6615	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
18		df-comf 16155	. . . 4 ⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
19		fvex 6113	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
20	3, 19	eqeltri 2684	. . . . . 6 ⊢ 𝐵 ∈ V
21	20, 20	xpex 6860	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
22	21, 20	mpt2ex 7136	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) ∈ V
23	17, 18, 22	fvmpt 6191	. . 3 ⊢ (𝐶 ∈ V → (compf‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
24		fvprc 6097	. . . 4 ⊢ (¬ 𝐶 ∈ V → (compf‘𝐶) = ∅)
25		fvprc 6097	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
26	3, 25	syl5eq 2656	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅)
27	26	xpeq2d 5063	. . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (𝐵 × 𝐵) = (𝐵 × ∅))
28		xp0 5471	. . . . . . 7 ⊢ (𝐵 × ∅) = ∅
29	27, 28	syl6eq 2660	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (𝐵 × 𝐵) = ∅)
30		mpt2eq12 6613	. . . . . 6 ⊢ (((𝐵 × 𝐵) = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
31	29, 26, 30	syl2anc 691	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
32		mpt20 6623	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅
33	31, 32	syl6eq 2660	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
34	24, 33	eqtr4d 2647	. . 3 ⊢ (¬ 𝐶 ∈ V → (compf‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
35	23, 34	pm2.61i 175	. 2 ⊢ (compf‘𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
36	1, 35	eqtri 2632	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  2^nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  comp_fccomf 16151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-comf 16155

This theorem is referenced by:  comffval  16182  comfffval2  16184  comfffn  16187  comfeq  16189