Theorem curf12 16867

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 16865	. . 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		fvex 6201	. . . . . 6 ⊢ (Base‘𝐷) ∈ V
13	6, 12	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
14	13	mptex 6486	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V
15	13, 13	mpt2ex 7247	. . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
16	14, 15	op2ndd 7179	. . 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))))
17	11, 16	syl 17	. 2 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
18		curf11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
19		curf12.y	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
20	19	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
21		ovex 6678	. . . . 5 ⊢ (𝑦𝐽𝑧) ∈ V
22	21	mptex 6486	. . . 4 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
23	22	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
24		curf12.g	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))
25	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
26		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑦 = 𝑌)
27		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
28	26, 27	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
29	25, 28	eleqtrrd 2704	. . . 4 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
30		ovexd 6680	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
31		simplrl 800	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
32	31	opeq2d 4409	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
33		simplrr 801	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
34	33	opeq2d 4409	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
35	32, 34	oveq12d 6668	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩))
36		eqidd 2623	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1 ‘𝑋) = ( 1 ‘𝑋))
37		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
38	35, 36, 37	oveq123d 6671	. . . 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))
39	29, 30, 38	fvmptdv2 6298	. . 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)))
40	18, 20, 23, 39	ovmpt2dv 6793	. 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)))
41	17, 40	mpd 15	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Basecbs 15857  Hom chom 15952  Catccat 16325  Idccid 16326   Func cfunc 16514   ×_c cxpc 16808   curry_F ccurf 16850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-curf 16854

This theorem is referenced by:  curf1cl  16868  curf2cl  16871  uncfcurf  16879  diag12  16884  yon12  16905