Theorem funfvima3 7094

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		ssel 3910	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
2		funfvop 6909	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
3	1, 2	impel 505	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)
4		sneq 4568	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	4	imaeq2d 5958	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
6	5	eleq2d 2824	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))
7		opeq1 4801	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ⟨𝑥, (𝐹‘𝐴)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
8	7	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝐴 → (⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
9		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
10		fvex 6769	. . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V
11	9, 10	elimasn 5986	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺)
12	6, 8, 11	vtoclbg 3497	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
13	12	ad2antll 725	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
14	3, 13	mpbird 256	. . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))
15	14	exp32 420	. 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))))
16	15	impcom 407	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  {csn 4558  ⟨cop 4564  dom cdm 5580   “ cima 5583  Fun wfun 6412  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  dfac3  9808