Theorem curry2val 8042

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 8040	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
3	2	fveq1d 6824	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))
5	4	fvmptndm 6961	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
6	5	adantl 481	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7		fndm 6585	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8		simpl 482	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐷 ∈ 𝐴)
9	8	con3i 154	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
10		ndmovg 7532	. . . . . . 7 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐷𝐹𝐶) = ∅)
11	7, 9, 10	syl2an 596	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → (𝐷𝐹𝐶) = ∅)
12	6, 11	eqtr4d 2767	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
13	12	ex 412	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
14	13	adantr 480	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15		oveq1 7356	. . . 4 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16		ovex 7382	. . . 4 ⊢ (𝐷𝐹𝐶) ∈ V
17	15, 4, 16	fvmpt 6930	. . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
18	14, 17	pm2.61d2 181	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
19	3, 18	eqtrd 2764	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284  {csn 4577   ↦ cmpt 5173   × cxp 5617  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   ∘ ccom 5623   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-1st 7924  df-2nd 7925

This theorem is referenced by:  curry2ima  32652