Theorem funfvima3 5887

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 5759	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
2		ssel 3221	. . . . . 6 ⊢ (𝐹 ⊆ 𝐺 → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
3	1, 2	syl5 32	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
4	3	imp 124	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)
5		simpr 110	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom 𝐹)
6		sneq 3680	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
7	6	imaeq2d 5076	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
8	7	eleq2d 2301	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))
9		opeq1 3862	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, (𝐹‘𝐴)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
10	9	eleq1d 2300	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
11	8, 10	bibi12d 235	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺) ↔ ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)))
12	11	adantl 277	. . . . . 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 2805	. . . . . . 7 ⊢ 𝑥 ∈ V
14		funfvex 5656	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V)
15		elimasng 5104	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ (𝐹‘𝐴) ∈ V) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺))
16	13, 14, 15	sylancr 414	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺))
17	5, 12, 16	vtocld 2856	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
18	17	adantl 277	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
19	4, 18	mpbird 167	. . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))
20	19	exp32 365	. 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))))
21	20	impcom 125	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  {csn 3669  ⟨cop 3672  dom cdm 4725   “ cima 4728  Fun wfun 5320  ‘cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)