curf11 - Metamath Proof Explorer

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Theorem curf11 18283

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 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18282	. . 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 6897	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7223	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8077	. . . 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 7997	. . 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 18	. 2 ⊢ (𝜑 → (1^st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)))
17		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	oveq2d 7428	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1^st ‘𝐹)𝑦) = (𝑋(1^st ‘𝐹)𝑌))
19		curf11.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20		ovexd 7447	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) ∈ V)
21	16, 18, 19, 20	fvmptd 6999	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = (𝑋(1^st ‘𝐹)𝑌))

Colors of variables: wff setvar class

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 curfcl 18289 uncfcurf 18296 diag11 18300 yon11 18321 tposcurf11 50058

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