Theorem funfvima3 7021

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 3880	. . . . 5 ⊢ (𝐹 ⊆ 𝐺 → (⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺))
2		funfvop 6839	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
3	1, 2	impel 509	. . . 4 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐺)
4		sneq 4536	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
5	4	imaeq2d 5913	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
6	5	eleq2d 2819	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))
7		opeq1 4769	. . . . . . 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 3404	. . . . . . 7 ⊢ 𝑥 ∈ V
10		fvex 6699	. . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V
11	9, 10	elimasn 5938	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹‘𝐴)⟩ ∈ 𝐺)
12	6, 8, 11	vtoclbg 3475	. . . . 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 260	. . 3 ⊢ ((𝐹 ⊆ 𝐺 ∧ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹)) → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))
15	14	exp32 424	. 2 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴}))))
16	15	impcom 411	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ⊆ wss 3853  {csn 4526  ⟨cop 4532  dom cdm 5535   “ cima 5538  Fun wfun 6343  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by:  dfac3  9633