Theorem curf2 18299

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 2740	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9		eqid 2740	. . . . 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 18293	. . . 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 6934	. . . . . 6 ⊢ 𝐴 ∈ V
14	13	mptex 7260	. . . . 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 8120	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧))))) ∈ V
16	14, 15	op2ndd 8041	. . . 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 7481	. . . . . 6 ⊢ (𝑥𝐻𝑦) ∈ V
22	21	mptex 7260	. . . . 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 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
27		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
28	26, 27	oveq12d 7466	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
29	25, 28	eleqtrrd 2847	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
30	7	fvexi 6934	. . . . . . 7 ⊢ 𝐵 ∈ V
31	30	mptex 7260	. . . . . 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 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑥 = 𝑋)
34	33	opeq1d 4903	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
35		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
36	35	opeq1d 4903	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
37	34, 36	oveq12d 7466	. . . . . . 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 2741	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼‘𝑧) = (𝐼‘𝑧))
40	37, 38, 39	oveq123d 7469	. . . . . 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 5268	. . . . 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 7047	. . . 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 7607	. . 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 2792	1 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  Basecbs 17258  Hom chom 17322  Catccat 17722  Idccid 17723   Func cfunc 17918   ×_c cxpc 18237   curry_F ccurf 18280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-curf 18284

This theorem is referenced by:  curf2val  18300  curf2cl  18301  curfcl  18302  diag2  18315  curf2ndf  18317