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Theorem funfvima3 5542
 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 funfvop 5425 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 ssel 3020 . . . . . 6 (𝐹𝐺 → (⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
31, 2syl5 32 . . . . 5 (𝐹𝐺 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
43imp 123 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)
5 simpr 109 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom 𝐹)
6 sneq 3461 . . . . . . . . . 10 (𝑥 = 𝐴 → {𝑥} = {𝐴})
76imaeq2d 4787 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
87eleq2d 2158 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
9 opeq1 3628 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, (𝐹𝐴)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
109eleq1d 2157 . . . . . . . 8 (𝑥 = 𝐴 → (⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
118, 10bibi12d 234 . . . . . . 7 (𝑥 = 𝐴 → (((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺) ↔ ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)))
1211adantl 272 . . . . . 6 (((Fun 𝐹𝐴 ∈ dom 𝐹) ∧ 𝑥 = 𝐴) → (((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺) ↔ ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)))
13 vex 2623 . . . . . . 7 𝑥 ∈ V
14 funfvex 5335 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
15 elimasng 4813 . . . . . . 7 ((𝑥 ∈ V ∧ (𝐹𝐴) ∈ V) → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺))
1613, 14, 15sylancr 406 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺))
175, 12, 16vtocld 2672 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
1817adantl 272 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
194, 18mpbird 166 . . 3 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))
2019exp32 358 . 2 (𝐹𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))))
2120impcom 124 1 ((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290   ∈ wcel 1439  Vcvv 2620   ⊆ wss 3000  {csn 3450  ⟨cop 3453  dom cdm 4451   “ cima 4454  Fun wfun 5022  ‘cfv 5028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036 This theorem is referenced by: (None)
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