Theorem curf2 18186

Description: Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curf2.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curf2.a	⊢ 𝐴 = (Base‘𝐶)
curf2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curf2.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curf2.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curf2.b	⊢ 𝐵 = (Base‘𝐷)
curf2.h	⊢ 𝐻 = (Hom ‘𝐶)
curf2.i	⊢ 𝐼 = (Id‘𝐷)
curf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
curf2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
curf2.l	⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)

Assertion

Ref	Expression
curf2	⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))

Distinct variable groups:   𝑧,𝐶   𝑧,𝐹   𝑧,𝐻   𝑧,𝐿   𝑧,𝐸   𝑧,𝐺   𝑧,𝐼   𝜑,𝑧   𝑧,𝐵   𝑧,𝐷   𝑧,𝑋   𝑧,𝐾   𝑧,𝑌

Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem curf2

Dummy variables 𝑥 𝑦 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf2.l	. 2 ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)
2		curf2.g	. . . . 5 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3		curf2.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
4		curf2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		curf2.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		curf2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7		curf2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
8		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2737	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
10		curf2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
11		curf2.i	. . . . 5 ⊢ 𝐼 = (Id‘𝐷)
12	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	curfval 18180	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
13	3	fvexi 6848	. . . . . 6 ⊢ 𝐴 ∈ V
14	13	mptex 7171	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
15	13, 13	mpoex 8025	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))) ∈ V
16	14, 15	op2ndd 7946	. . . 4 ⊢ (𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩ → (2nd ‘𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))))
17	12, 16	syl 17	. . 3 ⊢ (𝜑 → (2nd ‘𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))))
18		curf2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19		curf2.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐴)
21		ovex 7393	. . . . . 6 ⊢ (𝑥𝐻𝑦) ∈ V
22	21	mptex 7171	. . . . 5 ⊢ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))) ∈ V
23	22	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))) ∈ V)
24		curf2.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝐾 ∈ (𝑋𝐻𝑌))
26		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
27		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
28	26, 27	oveq12d 7378	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
29	25, 28	eleqtrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
30	7	fvexi 6848	. . . . . . 7 ⊢ 𝐵 ∈ V
31	30	mptex 7171	. . . . . 6 ⊢ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))) ∈ V
32	31	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))) ∈ V)
33		simplrl 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑥 = 𝑋)
34	33	opeq1d 4823	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
35		simplrr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
36	35	opeq1d 4823	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
37	34, 36	oveq12d 7378	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩) = (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩))
38		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑔 = 𝐾)
39		eqidd 2738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼‘𝑧) = (𝐼‘𝑧))
40	37, 38, 39	oveq123d 7381	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)) = (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))
41	40	mpteq2dv 5180	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))) = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
42	29, 32, 41	fvmptdv2 6960	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑋(2nd ‘𝐺)𝑌) = (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))) → ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))))
43	18, 20, 23, 42	ovmpodv 7517	. . 3 ⊢ (𝜑 → ((2nd ‘𝐺) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))) → ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)))))
44	17, 43	mpd 15	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
45	1, 44	eqtrid 2784	1 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222  Catccat 17621  Idccid 17622   Func cfunc 17812   ×_c cxpc 18125   curry_F ccurf 18167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-curf 18171

This theorem is referenced by:  curf2val  18187  curf2cl  18188  curfcl  18189  diag2  18202  curf2ndf  18204  tposcurf2  49787