Theorem curry1f 7477

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		fovrn 7006	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
2	1	3expa 1147	. . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷)
3	2	fmpttd 6579	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)):𝐵⟶𝐷)
4		ffn 6225	. . . 4 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵))
5		curry1.1	. . . . 5 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
6	5	curry1 7475	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
7	4, 6	sylan 575	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
8	7	feq1d 6210	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → (𝐺:𝐵⟶𝐷 ↔ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)):𝐵⟶𝐷))
9	3, 8	mpbird 248	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  Vcvv 3350  {csn 4336   ↦ cmpt 4890   × cxp 5277  ^◡ccnv 5278   ↾ cres 5281   ∘ ccom 5283   Fn wfn 6065  ⟶wf 6066  (class class class)co 6846  2^nd c2nd 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-1st 7370  df-2nd 7371

This theorem is referenced by:  nvinvfval  27972