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Theorem funfvima3 7256
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 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))

Proof of Theorem funfvima3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3977 . . . . 5 (𝐹𝐺 → (⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
2 funfvop 7070 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
31, 2impel 505 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)
4 sneq 4636 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
54imaeq2d 6078 . . . . . . 7 (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
65eleq2d 2827 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
7 opeq1 4873 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, (𝐹𝐴)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87eleq1d 2826 . . . . . 6 (𝑥 = 𝐴 → (⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
9 vex 3484 . . . . . . 7 𝑥 ∈ V
10 fvex 6919 . . . . . . 7 (𝐹𝐴) ∈ V
119, 10elimasn 6108 . . . . . 6 ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺)
126, 8, 11vtoclbg 3557 . . . . 5 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
1312ad2antll 729 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
143, 13mpbird 257 . . 3 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))
1514exp32 420 . 2 (𝐹𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))))
1615impcom 407 1 ((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  {csn 4626  cop 4632  dom cdm 5685  cima 5688  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  dfac3  10161
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