Theorem funfvima3 5313

Description: A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)

Assertion

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

Proof of Theorem funfvima3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvop 5200	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ⟨A, (𝐹‘A)⟩ ∈ 𝐹)
2		ssel 2912	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → (⟨A, (𝐹‘A)⟩ ∈ 𝐹 → ⟨A, (𝐹‘A)⟩ ∈ 𝐺))
3	1, 2	syl5 28	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ⟨A, (𝐹‘A)⟩ ∈ 𝐺))
4	3	imp 115	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ A ∈ dom 𝐹)) → ⟨A, (𝐹‘A)⟩ ∈ 𝐺)
5		ax-ia2 100	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → A ∈ dom 𝐹)
6		sneq 3357	. . . . . . . . . 10 ⊢ (x = A → {x} = {A})
7	6	imaeq2d 4591	. . . . . . . . 9 ⊢ (x = A → (𝐺 “ {x}) = (𝐺 “ {A}))
8	7	eleq2d 2085	. . . . . . . 8 ⊢ (x = A → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ (𝐹‘A) ∈ (𝐺 “ {A})))
9		opeq1 3519	. . . . . . . . 9 ⊢ (x = A → ⟨x, (𝐹‘A)⟩ = ⟨A, (𝐹‘A)⟩)
10	9	eleq1d 2084	. . . . . . . 8 ⊢ (x = A → (⟨x, (𝐹‘A)⟩ ∈ 𝐺 ↔ ⟨A, (𝐹‘A)⟩ ∈ 𝐺))
11	8, 10	bibi12d 224	. . . . . . 7 ⊢ (x = A → (((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺) ↔ ((𝐹‘A) ∈ (𝐺 “ {A}) ↔ ⟨A, (𝐹‘A)⟩ ∈ 𝐺)))
12	11	adantl 262	. . . . . 6 ⊢ (((Fun 𝐹 ∧ A ∈ dom 𝐹) ∧ x = A) → (((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺) ↔ ((𝐹‘A) ∈ (𝐺 “ {A}) ↔ ⟨A, (𝐹‘A)⟩ ∈ 𝐺)))
13		vex 2534	. . . . . . 7 ⊢ x ∈ V
14		funfvex 5113	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → (𝐹‘A) ∈ V)
15		elimasng 4616	. . . . . . 7 ⊢ ((x ∈ V ∧ (𝐹‘A) ∈ V) → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺))
16	13, 14, 15	sylancr 395	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺))
17	5, 12, 16	vtocld 2579	. . . . 5 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ((𝐹‘A) ∈ (𝐺 “ {A}) ↔ ⟨A, (𝐹‘A)⟩ ∈ 𝐺))
18	17	adantl 262	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ A ∈ dom 𝐹)) → ((𝐹‘A) ∈ (𝐺 “ {A}) ↔ ⟨A, (𝐹‘A)⟩ ∈ 𝐺))
19	4, 18	mpbird 156	. . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ A ∈ dom 𝐹)) → (𝐹‘A) ∈ (𝐺 “ {A}))
20	19	exp32 347	. 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (A ∈ dom 𝐹 → (𝐹‘A) ∈ (𝐺 “ {A}))))
21	20	impcom 116	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (A ∈ dom 𝐹 → (𝐹‘A) ∈ (𝐺 “ {A})))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  Vcvv 2531   ⊆ wss 2890  {csn 3346  ⟨cop 3349  dom cdm 4268   “ cima 4271  Fun wfun 4819  ‘cfv 4825

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833

This theorem is referenced by: (None)