Theorem curry2val 7660

Description: The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
curry2.1	⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))

Assertion

Ref	Expression
curry2val	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))

Proof of Theorem curry2val

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry2.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
2	1	curry2 7658	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
3	2	fveq1d 6540	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4		eqid 2795	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))
5	4	fvmptndm 6663	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
6	5	adantl 482	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7		fndm 6325	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8		simpl 483	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐷 ∈ 𝐴)
9	8	con3i 157	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
10		ndmovg 7187	. . . . . . 7 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐷𝐹𝐶) = ∅)
11	7, 9, 10	syl2an 595	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → (𝐷𝐹𝐶) = ∅)
12	6, 11	eqtr4d 2834	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
13	12	ex 413	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
14	13	adantr 481	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15		oveq1 7023	. . . 4 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16		ovex 7048	. . . 4 ⊢ (𝐷𝐹𝐶) ∈ V
17	15, 4, 16	fvmpt 6635	. . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
18	14, 17	pm2.61d2 182	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
19	3, 18	eqtrd 2831	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437  ∅c0 4211  {csn 4472   ↦ cmpt 5041   × cxp 5441  ^◡ccnv 5442  dom cdm 5443   ↾ cres 5445   ∘ ccom 5447   Fn wfn 6220  ‘cfv 6225  (class class class)co 7016  1^st c1st 7543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-1st 7545  df-2nd 7546

This theorem is referenced by:  curry2ima  30132