Theorem funfvima3 3868

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 2072	. . . . . 6 ⊢ (F ⊆ G → (⟨A, (F ‘A)⟩ ∈ F → ⟨A, (F ‘A)⟩ ∈ G))
2		funfvop 3817	. . . . . 6 ⊢ ((Fun F ⋀ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ F)
3	1, 2	syl5 21	. . . . 5 ⊢ (F ⊆ G → ((Fun F ⋀ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ G))
4	3	imp 350	. . . 4 ⊢ ((F ⊆ G ⋀ (Fun F ⋀ A ∈ dom F)) → ⟨A, (F ‘A)⟩ ∈ G)
5		sneq 2427	. . . . . . . 8 ⊢ (x = A → {x} = {A})
6	5	imaeq2d 3418	. . . . . . 7 ⊢ (x = A → (G “ {x}) = (G “ {A}))
7	6	eleq2d 1548	. . . . . 6 ⊢ (x = A → ((F ‘A) ∈ (G “ {x}) ↔ (F ‘A) ∈ (G “ {A})))
8		opeq1 2499	. . . . . . 7 ⊢ (x = A → ⟨x, (F ‘A)⟩ = ⟨A, (F ‘A)⟩)
9	8	eleq1d 1547	. . . . . 6 ⊢ (x = A → (⟨x, (F ‘A)⟩ ∈ G ↔ ⟨A, (F ‘A)⟩ ∈ G))
10		visset 1820	. . . . . . 7 ⊢ x ∈ V
11		fvex 3746	. . . . . . 7 ⊢ (F ‘A) ∈ V
12	10, 11	elimasn 3440	. . . . . 6 ⊢ ((F ‘A) ∈ (G “ {x}) ↔ ⟨x, (F ‘A)⟩ ∈ G)
13	7, 9, 12	vtoclbg 1855	. . . . 5 ⊢ (A ∈ dom F → ((F ‘A) ∈ (G “ {A}) ↔ ⟨A, (F ‘A)⟩ ∈ G))
14	13	ad2antll 409	. . . 4 ⊢ ((F ⊆ G ⋀ (Fun F ⋀ A ∈ dom F)) → ((F ‘A) ∈ (G “ {A}) ↔ ⟨A, (F ‘A)⟩ ∈ G))
15	4, 14	mpbird 196	. . 3 ⊢ ((F ⊆ G ⋀ (Fun F ⋀ A ∈ dom F)) → (F ‘A) ∈ (G “ {A}))
16	15	exp32 379	. 2 ⊢ (F ⊆ G → (Fun F → (A ∈ dom F → (F ‘A) ∈ (G “ {A}))))
17	16	impcom 351	1 ⊢ ((Fun F ⋀ F ⊆ G) → (A ∈ dom F → (F ‘A) ∈ (G “ {A})))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 960   ∈ wcel 962   ⊆ wss 2056  {csn 2419  ⟨cop 2421  dom cdm 3184   “ cima 3187  Fun wfun 3190   ‘cfv 3196

This theorem is referenced by:  aceq3 4745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-fv 3212

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