Theorem curry2val 8059

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 8057	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
3	2	fveq1d 6842	. 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 6979	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
6	5	adantl 481	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅)
7		fndm 6601	. . . . . . 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 7550	. . . . . . 7 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐷𝐹𝐶) = ∅)
11	7, 9, 10	syl2an 597	. . . . . 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 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 7374	. . . 4 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶))
16		ovex 7400	. . . 4 ⊢ (𝐷𝐹𝐶) ∈ V
17	15, 4, 16	fvmpt 6947	. . 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 2771	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  {csn 4567   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   ∘ ccom 5635   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940

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-sep 5231  ax-nul 5241  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-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-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-1st 7942  df-2nd 7943

This theorem is referenced by:  curry2ima  32782