Theorem curry1val 4084

Description: The value of a curried function with a constant first argument.

Hypothesis

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

Assertion

Ref	Expression
curry1val	⊢ ((F Fn (A × B) ⋀ C ∈ A ⋀ D ∈ U) → (G ‘D) = (CFD))

Proof of Theorem curry1val

Step	Hyp	Ref	Expression
1		curry1.1	. . . . 5 ⊢ G = (F ∘ ◡(2nd ↾ ({C} × V)))
2	1	curry1 4082	. . . 4 ⊢ ((F Fn (A × B) ⋀ C ∈ A) → G = {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))})
3	2	fveq1d 3711	. . 3 ⊢ ((F Fn (A × B) ⋀ C ∈ A) → (G ‘D) = ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D))
4	3	3adant3 797	. 2 ⊢ ((F Fn (A × B) ⋀ C ∈ A ⋀ D ∈ U) → (G ‘D) = ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D))
5		eqid 1468	. . . . . . . 8 ⊢ {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} = {⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))}
6	5	fvopab4ndm 3769	. . . . . . 7 ⊢ (¬ D ∈ B → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = ∅)
7	6	3ad2ant3 800	. . . . . 6 ⊢ ((F Fn (A × B) ⋀ D ∈ U ⋀ ¬ D ∈ B) → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = ∅)
8		ndmoprg 4028	. . . . . . 7 ⊢ ((dom F = (A × B) ⋀ D ∈ U ⋀ ¬ (C ∈ A ⋀ D ∈ B)) → (CFD) = ∅)
9		fndm 3573	. . . . . . 7 ⊢ (F Fn (A × B) → dom F = (A × B))
10		id 59	. . . . . . 7 ⊢ (D ∈ U → D ∈ U)
11		pm3.27 323	. . . . . . . 8 ⊢ ((C ∈ A ⋀ D ∈ B) → D ∈ B)
12	11	con3i 98	. . . . . . 7 ⊢ (¬ D ∈ B → ¬ (C ∈ A ⋀ D ∈ B))
13	8, 9, 10, 12	syl3an 866	. . . . . 6 ⊢ ((F Fn (A × B) ⋀ D ∈ U ⋀ ¬ D ∈ B) → (CFD) = ∅)
14	7, 13	eqtr4d 1502	. . . . 5 ⊢ ((F Fn (A × B) ⋀ D ∈ U ⋀ ¬ D ∈ B) → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = (CFD))
15	14	3expia 833	. . . 4 ⊢ ((F Fn (A × B) ⋀ D ∈ U) → (¬ D ∈ B → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = (CFD)))
16		opreq2 3954	. . . . 5 ⊢ (x = D → (CFx) = (CFD))
17		oprex 3968	. . . . 5 ⊢ (CFD) ∈ V
18	16, 5, 17	fvopab4 3765	. . . 4 ⊢ (D ∈ B → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = (CFD))
19	15, 18	pm2.61d2 129	. . 3 ⊢ ((F Fn (A × B) ⋀ D ∈ U) → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = (CFD))
20	19	3adant2 796	. 2 ⊢ ((F Fn (A × B) ⋀ C ∈ A ⋀ D ∈ U) → ({⟨x, y⟩∣(x ∈ B ⋀ y = (CFx))} ‘D) = (CFD))
21	4, 20	eqtrd 1499	1 ⊢ ((F Fn (A × B) ⋀ C ∈ A ⋀ D ∈ U) → (G ‘D) = (CFD))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955  Vcvv 1802  ∅c0 2270  {csn 2399  {copab 2656   × cxp 3158  ^◡ccnv 3159  dom cdm 3160   ↾ cres 3162   ∘ ccom 3164   Fn wfn 3167   ‘cfv 3172  (class class class)co 3948  2^nd c2nd 4062

This theorem is referenced by:  invfval 8201  hhssabl 9053

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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