Theorem funfvima3 5335

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 5222	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ⟨A, (𝐹‘A)⟩ ∈ 𝐹)
2		ssel 2933	. . . . . 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		simpr 103	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → A ∈ dom 𝐹)
6		sneq 3378	. . . . . . . . . 10 ⊢ (x = A → {x} = {A})
7	6	imaeq2d 4611	. . . . . . . . 9 ⊢ (x = A → (𝐺 “ {x}) = (𝐺 “ {A}))
8	7	eleq2d 2104	. . . . . . . 8 ⊢ (x = A → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ (𝐹‘A) ∈ (𝐺 “ {A})))
9		opeq1 3540	. . . . . . . . 9 ⊢ (x = A → ⟨x, (𝐹‘A)⟩ = ⟨A, (𝐹‘A)⟩)
10	9	eleq1d 2103	. . . . . . . 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 2554	. . . . . . 7 ⊢ x ∈ V
14		funfvex 5135	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → (𝐹‘A) ∈ V)
15		elimasng 4636	. . . . . . 7 ⊢ ((x ∈ V ∧ (𝐹‘A) ∈ V) → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺))
16	13, 14, 15	sylancr 393	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ((𝐹‘A) ∈ (𝐺 “ {x}) ↔ ⟨x, (𝐹‘A)⟩ ∈ 𝐺))
17	5, 12, 16	vtocld 2600	. . . . 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 1242   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  {csn 3367  ⟨cop 3370  dom cdm 4288   “ cima 4291  Fun wfun 4839  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)