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Theorem curf11 18192

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 18191	. . 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 6854	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7178	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1^st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8032	. . . 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 7952	. . 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 7383	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑋(1^st ‘𝐹)𝑦) = (𝑋(1^st ‘𝐹)𝑌))
19		curf11.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20		ovexd 7402	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐹)𝑌) ∈ V)
21	16, 18, 19, 20	fvmptd 6955	1 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = (𝑋(1^st ‘𝐹)𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 1^st c1st 7940 2^nd c2nd 7941 Basecbs 17179 Hom chom 17231 Catccat 17630 Idccid 17631 Func cfunc 17821 ×_c cxpc 18134 curry_F ccurf 18176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-curf 18180

This theorem is referenced by: curf1cl 18194 curf2cl 18197 curfcl 18198 uncfcurf 18205 diag11 18209 yon11 18230 tposcurf11 49772

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