Theorem curry2f 8090

Description: Functionality 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
curry2f	⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷)

Proof of Theorem curry2f

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6691	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵))
2		curry2.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
3	2	curry2 8089	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
4	1, 3	sylan 580	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)))
5		fovcdm 7562	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝑥𝐹𝐶) ∈ 𝐷)
6	5	3com23 1126	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
7	6	3expa 1118	. 2 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷)
8	4, 7	fmpt3d 7091	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  (class class class)co 7390  1^st c1st 7969

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-1st 7971  df-2nd 7972

This theorem is referenced by:  curry2ima  32639