Theorem mpocurryvald 8267

Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)

Hypotheses

Ref	Expression
mpocurryd.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
mpocurryd.c	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
mpocurryd.n	⊢ (𝜑 → 𝑌 ≠ ∅)
mpocurryvald.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)
mpocurryvald.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
mpocurryvald	⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mpocurryvald

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpocurryd.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
2		mpocurryd.c	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
3		mpocurryd.n	. . . 4 ⊢ (𝜑 → 𝑌 ≠ ∅)
4	1, 2, 3	mpocurryd 8266	. . 3 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
5		nfcv 2898	. . . 4 ⊢ Ⅎ𝑎(𝑦 ∈ 𝑌 ↦ 𝐶)
6		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑌
7		nfcsb1v 3914	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
8	6, 7	nfmpt 5249	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)
9		csbeq1a 3903	. . . . 5 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
10	9	mpteq2dv 5244	. . . 4 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))
11	5, 8, 10	cbvmpt 5253	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))
12	4, 11	eqtrdi 2783	. 2 ⊢ (𝜑 → curry 𝐹 = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)))
13		csbeq1 3892	. . . 4 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
14	13	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
15	14	mpteq2dv 5244	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶))
16		mpocurryvald.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
17		mpocurryvald.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
18	17	mptexd 7230	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶) ∈ V)
19	12, 15, 16, 18	fvmptd 7006	1 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  Vcvv 3469  ⦋csb 3889  ∅c0 4318   ↦ cmpt 5225  ‘cfv 6542   ∈ cmpo 7416  curry ccur 8262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-cur 8264

This theorem is referenced by:  fvmpocurryd  8268  pmatcollpw3lem  22659  logbmpt  26694