Theorem funfvima3 2906

Description: A class including a function contains the function's value in the image of the singleton of the argument.

Assertion

Ref	Expression
funfvima3	⊢ ((Fun F ∧ F ⊆ G) → (A ∈ dom F → (F ‘A) ∈ (G “ {A})))

Proof of Theorem funfvima3

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . 7 ⊢ (F ⊆ G → (⟨A, (F ‘A)⟩ ∈ F → ⟨A, (F ‘A)⟩ ∈ G))
2		funopfv 2886	. . . . . . 7 ⊢ ((Fun F ∧ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ F)
3	1, 2	syl5 22	. . . . . 6 ⊢ (F ⊆ G → ((Fun F ∧ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ G))
4	3	imp 277	. . . . 5 ⊢ ((F ⊆ G ∧ (Fun F ∧ A ∈ dom F)) → ⟨A, (F ‘A)⟩ ∈ G)
5		sneq 1816	. . . . . . . . 9 ⊢ (x = A → {x} = {A})
6		imaeq2 2603	. . . . . . . . 9 ⊢ ({x} = {A} → (G “ {x}) = (G “ {A}))
7	5, 6	syl 12	. . . . . . . 8 ⊢ (x = A → (G “ {x}) = (G “ {A}))
8	7	eleq2d 1156	. . . . . . 7 ⊢ (x = A → ((F ‘A) ∈ (G “ {x}) ↔ (F ‘A) ∈ (G “ {A})))
9		opeq1 1876	. . . . . . . 8 ⊢ (x = A → ⟨x, (F ‘A)⟩ = ⟨A, (F ‘A)⟩)
10	9	eleq1d 1155	. . . . . . 7 ⊢ (x = A → (⟨x, (F ‘A)⟩ ∈ G ↔ ⟨A, (F ‘A)⟩ ∈ G))
11		visset 1350	. . . . . . . 8 ⊢ x ∈ V
12		fvex 2838	. . . . . . . 8 ⊢ (F ‘A) ∈ V
13	11, 12	elimasn 2617	. . . . . . 7 ⊢ ((F ‘A) ∈ (G “ {x}) ↔ ⟨x, (F ‘A)⟩ ∈ G)
14	8, 10, 13	vtoclbg 1384	. . . . . 6 ⊢ (A ∈ dom F → ((F ‘A) ∈ (G “ {A}) ↔ ⟨A, (F ‘A)⟩ ∈ G))
15	14	ad2antrr 323	. . . . 5 ⊢ ((F ⊆ G ∧ (Fun F ∧ A ∈ dom F)) → ((F ‘A) ∈ (G “ {A}) ↔ ⟨A, (F ‘A)⟩ ∈ G))
16	4, 15	mpbird 171	. . . 4 ⊢ ((F ⊆ G ∧ (Fun F ∧ A ∈ dom F)) → (F ‘A) ∈ (G “ {A}))
17	16	exp32 294	. . 3 ⊢ (F ⊆ G → (Fun F → (A ∈ dom F → (F ‘A) ∈ (G “ {A}))))
18	17	com12 13	. 2 ⊢ (Fun F → (F ⊆ G → (A ∈ dom F → (F ‘A) ∈ (G “ {A}))))
19	18	imp 277	1 ⊢ ((Fun F ∧ F ⊆ G) → (A ∈ dom F → (F ‘A) ∈ (G “ {A})))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  {csn 1808  ⟨cop 1810  dom cdm 2410   “ cima 2413  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  aceq3 3556

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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