Theorem curf12 18284

Description: The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a	⊢ 𝐴 = (Base‘𝐶)
curfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfval.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b	⊢ 𝐵 = (Base‘𝐷)
curf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf1.k	⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
curf11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
curf12.j	⊢ 𝐽 = (Hom ‘𝐷)
curf12.1	⊢ 1 = (Id‘𝐶)
curf12.y	⊢ (𝜑 → 𝑍 ∈ 𝐵)
curf12.g	⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))

Assertion

Ref	Expression
curf12	⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))

Proof of Theorem curf12

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

Step	Hyp	Ref	Expression
1		curfval.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		curfval.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
3		curfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		curfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
7		curf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
8		curf1.k	. . . 4 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
9		curf12.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
10		curf12.1	. . . 4 ⊢ 1 = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18282	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12	6	fvexi 6897	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7223	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8077	. . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
15	13, 14	op2ndd 7998	. . 3 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
16	11, 15	syl 18	. 2 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
17		curf11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18		curf12.y	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	18	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
20		ovex 7445	. . . . 5 ⊢ (𝑦𝐽𝑧) ∈ V
21	20	mptex 7223	. . . 4 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
23		curf12.g	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))
24	23	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑦 = 𝑌)
26		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
27	25, 26	oveq12d 7430	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
28	24, 27	eleqtrrd 2866	. . . 4 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29		ovexd 7447	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30		simplrl 788	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
31	30	opeq2d 4846	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32		simplrr 789	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
33	32	opeq2d 4846	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
34	31, 33	oveq12d 7430	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩))
35		eqidd 2764	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1 ‘𝑋) = ( 1 ‘𝑋))
36		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
37	34, 35, 36	oveq123d 7433	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))
38	28, 29, 37	fvmptdv2 7010	. . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → ((𝑌(2nd ‘𝐾)𝑍) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻)))
39	17, 19, 22, 38	ovmpodv 7569	. 2 ⊢ (𝜑 → ((2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻)))
40	16, 39	mpd 16	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322  Catccat 17721  Idccid 17722   Func cfunc 17912   ×_c cxpc 18225   curry_F ccurf 18267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-curf 18271

This theorem is referenced by:  curf1cl  18285  curf2cl  18288  uncfcurf  18296  diag12  18301  yon12  18322  tposcurf12  50053