Theorem curry1f 4083

Description: Functionality of a curried function with a constant first argument.

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

Step	Hyp	Ref	Expression
1		foprrn 4020	. . . . 5 ⊢ ((F:(A × B)–→D ⋀ C ∈ A ⋀ x ∈ B) → (CFx) ∈ D)
2	1	3expa 831	. . . 4 ⊢ (((F:(A × B)–→D ⋀ C ∈ A) ⋀ x ∈ B) → (CFx) ∈ D)
3	2	r19.21aiva 1706	. . 3 ⊢ ((F:(A × B)–→D ⋀ C ∈ A) → ∀x ∈ B (CFx) ∈ D)
4		eqid 1468	. . . 4 ⊢ {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} = {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))}
5	4	fopab2 3808	. . 3 ⊢ (∀x ∈ B (CFx) ∈ D ↔ {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))}:B–→D)
6	3, 5	sylib 198	. 2 ⊢ ((F:(A × B)–→D ⋀ C ∈ A) → {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))}:B–→D)
7		curry1.1	. . . . 5 ⊢ G = (F ∘ ◡(2nd ↾ ({C} × V)))
8	7	curry1 4082	. . . 4 ⊢ ((F Fn (A × B) ⋀ C ∈ A) → G = {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))})
9		ffn 3613	. . . 4 ⊢ (F:(A × B)–→D → F Fn (A × B))
10	8, 9	sylan 448	. . 3 ⊢ ((F:(A × B)–→D ⋀ C ∈ A) → G = {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))})
11	10	feq1d 3610	. 2 ⊢ ((F:(A × B)–→D ⋀ C ∈ A) → (G:B–→D ↔ {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))}:B–→D))
12	6, 11	mpbird 196	1 ⊢ ((F:(A × B)–→D ⋀ C ∈ A) → G:B–→D)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∀wral 1637  Vcvv 1802  {csn 2399  {copab 2656   × cxp 3158  ^◡ccnv 3159   ↾ cres 3162   ∘ ccom 3164   Fn wfn 3167  –→wf 3168  (class class class)co 3948  2^nd c2nd 4062

This theorem is referenced by:  invfval 8201

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-2nd 4064

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