curf11 - Metamath Proof Explorer

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Theorem curf11 18243

Description: Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

**Hypotheses**
Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curry_F 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 = ((1^st ‘𝐺)‘𝑋)
curf11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
curf11	⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = (𝑋(1^st ‘𝐹)𝑌))

Proof of Theorem curf11 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⟩ curry_F 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 = ((1^st ‘𝐺)‘𝑋)
9		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2736	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18242	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2^nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12	6	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7220	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8083	. . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2^nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
15	13, 14	op1std 8003	. . 3 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2^nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1^st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)))
16	11, 15	syl 17	. 2 ⊢ (𝜑 → (1^st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)))
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	oveq2d 7426	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1^st ‘𝐹)𝑦) = (𝑋(1^st ‘𝐹)𝑌))
19		curf11.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20		ovexd 7445	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) ∈ V)
21	16, 18, 19, 20	fvmptd 6998	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = (𝑋(1^st ‘𝐹)𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ⟨cop 4612 ↦ cmpt 5206 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17233 Hom chom 17287 Catccat 17681 Idccid 17682 Func cfunc 17872 ×_c cxpc 18185 curry_F ccurf 18227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-curf 18231

This theorem is referenced by: curf1cl 18245 curf2cl 18248 curfcl 18249 uncfcurf 18256 diag11 18260 yon11 18281 tposcurf11 49188

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