Theorem curry2val 7855

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 7853	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
3	2	fveq1d 6697	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷))
4		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))
5	4	fvmptndm 6826	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
6	5	adantl 485	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7		fndm 6459	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8		simpl 486	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐷 ∈ 𝐴)
9	8	con3i 157	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
10		ndmovg 7369	. . . . . . 7 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐷𝐹𝐶) = ∅)
11	7, 9, 10	syl2an 599	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → (𝐷𝐹𝐶) = ∅)
12	6, 11	eqtr4d 2774	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
13	12	ex 416	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
14	13	adantr 484	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)))
15		oveq1 7198	. . . 4 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16		ovex 7224	. . . 4 ⊢ (𝐷𝐹𝐶) ∈ V
17	15, 4, 16	fvmpt 6796	. . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
18	14, 17	pm2.61d2 184	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))
19	3, 18	eqtrd 2771	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  Vcvv 3398  ∅c0 4223  {csn 4527   ↦ cmpt 5120   × cxp 5534  ^◡ccnv 5535  dom cdm 5536   ↾ cres 5538   ∘ ccom 5540   Fn wfn 6353  ‘cfv 6358  (class class class)co 7191  1^st c1st 7737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-1st 7739  df-2nd 7740

This theorem is referenced by:  curry2ima  30715