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Theorem funfvima3 7240
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 3975 . . . . 5 (𝐹𝐺 → (⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
2 funfvop 7051 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
31, 2impel 506 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)
4 sneq 4638 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
54imaeq2d 6059 . . . . . . 7 (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
65eleq2d 2819 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
7 opeq1 4873 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, (𝐹𝐴)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87eleq1d 2818 . . . . . 6 (𝑥 = 𝐴 → (⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
9 vex 3478 . . . . . . 7 𝑥 ∈ V
10 fvex 6904 . . . . . . 7 (𝐹𝐴) ∈ V
119, 10elimasn 6088 . . . . . 6 ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺)
126, 8, 11vtoclbg 3559 . . . . 5 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
1312ad2antll 727 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
143, 13mpbird 256 . . 3 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))
1514exp32 421 . 2 (𝐹𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))))
1615impcom 408 1 ((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wss 3948  {csn 4628  cop 4634  dom cdm 5676  cima 5679  Fun wfun 6537  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  dfac3  10118
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