Theorem curf1fval 18239

Description: Value of the object part of the curry functor. (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)
curfval.j	⊢ 𝐽 = (Hom ‘𝐷)
curfval.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
curf1fval	⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))

Distinct variable groups:   𝑥,𝑔,𝑦,𝑧, 1   𝑥,𝐴,𝑦   𝐵,𝑔,𝑥,𝑦,𝑧   𝐶,𝑔,𝑥,𝑦,𝑧   𝐷,𝑔,𝑥,𝑦,𝑧   𝜑,𝑔,𝑥,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽,𝑥   𝑔,𝐹,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐸(𝑥)   𝐺(𝑥,𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curf1fval

Step	Hyp	Ref	Expression
1		curfval.g	. . 3 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		curfval.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
3		curfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curfval.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		curfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
7		curfval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
8		curfval.1	. . 3 ⊢ 1 = (Id‘𝐶)
9		eqid 2761	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2761	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curfval 18238	. 2 ⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
12	2	fvexi 6877	. . . 4 ⊢ 𝐴 ∈ V
13	12	mptex 7203	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
14	12, 12	mpoex 8056	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) ∈ V
15	13, 14	op1std 7976	. 2 ⊢ (𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
16	11, 15	syl 17	1 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ⟨cop 4587   ↦ cmpt 5180  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  1^st c1st 7964  2^nd c2nd 7965  Basecbs 17228  Hom chom 17280  Catccat 17679  Idccid 17680   Func cfunc 17870   ×_c cxpc 18183   curry_F ccurf 18225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-curf 18229

This theorem is referenced by:  curf1  18240