Theorem funfvima3 7179

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 3924	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
2		funfvop 6992	. . . . 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 4587	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	4	imaeq2d 6016	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
6	5	eleq2d 2819	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))
7		opeq1 4826	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ⟨𝑥, (𝐹‘𝐴)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
8	7	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝐴 → (⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
9		vex 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
10		fvex 6844	. . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V
11	9, 10	elimasn 6046	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺)
12	6, 8, 11	vtoclbg 3511	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
13	12	ad2antll 729	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ((𝐹‘𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
14	3, 13	mpbird 257	. . 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  {csn 4577  ⟨cop 4583  dom cdm 5621   “ cima 5624  Fun wfun 6483  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  dfac3  10023