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Theorem curf11 18194

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 2730	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2730	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18193	. . 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 6875	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7200	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8061	. . . 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 7981	. . 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 7406	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1^st ‘𝐹)𝑦) = (𝑋(1^st ‘𝐹)𝑌))
19		curf11.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20		ovexd 7425	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) ∈ V)
21	16, 18, 19, 20	fvmptd 6978	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 3450 ⟨cop 4598 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17186 Hom chom 17238 Catccat 17632 Idccid 17633 Func cfunc 17823 ×_c cxpc 18136 curry_F ccurf 18178

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-curf 18182

This theorem is referenced by: curf1cl 18196 curf2cl 18199 curfcl 18200 uncfcurf 18207 diag11 18211 yon11 18232 tposcurf11 49290

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