Theorem curry1f 8042

Description: Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1f	⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷)

Proof of Theorem curry1f

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6656	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵))
2		curry1.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
3	2	curry1 8040	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
4	1, 3	sylan 580	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
5		fovcdm 7522	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
6	5	3expa 1118	. 2 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
7	4, 6	fmpt3d 7055	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4575   ↦ cmpt 5174   × cxp 5617  ^◡ccnv 5618   ↾ cres 5621   ∘ ccom 5623   Fn wfn 6481  ⟶wf 6482  (class class class)co 7352  2^nd c2nd 7926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-1st 7927  df-2nd 7928

This theorem is referenced by:  nvinvfval  30622