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Theorem funfvima3 5335
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 𝐹𝐺) → (A dom 𝐹 → (𝐹A) (𝐺 “ {A})))

Proof of Theorem funfvima3
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 funfvop 5222 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐹)
2 ssel 2933 . . . . . 6 (𝐹𝐺 → (⟨A, (𝐹A)⟩ 𝐹 → ⟨A, (𝐹A)⟩ 𝐺))
31, 2syl5 28 . . . . 5 (𝐹𝐺 → ((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐺))
43imp 115 . . . 4 ((𝐹𝐺 (Fun 𝐹 A dom 𝐹)) → ⟨A, (𝐹A)⟩ 𝐺)
5 simpr 103 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → A dom 𝐹)
6 sneq 3378 . . . . . . . . . 10 (x = A → {x} = {A})
76imaeq2d 4611 . . . . . . . . 9 (x = A → (𝐺 “ {x}) = (𝐺 “ {A}))
87eleq2d 2104 . . . . . . . 8 (x = A → ((𝐹A) (𝐺 “ {x}) ↔ (𝐹A) (𝐺 “ {A})))
9 opeq1 3540 . . . . . . . . 9 (x = A → ⟨x, (𝐹A)⟩ = ⟨A, (𝐹A)⟩)
109eleq1d 2103 . . . . . . . 8 (x = A → (⟨x, (𝐹A)⟩ 𝐺 ↔ ⟨A, (𝐹A)⟩ 𝐺))
118, 10bibi12d 224 . . . . . . 7 (x = A → (((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺) ↔ ((𝐹A) (𝐺 “ {A}) ↔ ⟨A, (𝐹A)⟩ 𝐺)))
1211adantl 262 . . . . . 6 (((Fun 𝐹 A dom 𝐹) x = A) → (((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺) ↔ ((𝐹A) (𝐺 “ {A}) ↔ ⟨A, (𝐹A)⟩ 𝐺)))
13 vex 2554 . . . . . . 7 x V
14 funfvex 5135 . . . . . . 7 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
15 elimasng 4636 . . . . . . 7 ((x V (𝐹A) V) → ((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺))
1613, 14, 15sylancr 393 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺))
175, 12, 16vtocld 2600 . . . . 5 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) (𝐺 “ {A}) ↔ ⟨A, (𝐹A)⟩ 𝐺))
1817adantl 262 . . . 4 ((𝐹𝐺 (Fun 𝐹 A dom 𝐹)) → ((𝐹A) (𝐺 “ {A}) ↔ ⟨A, (𝐹A)⟩ 𝐺))
194, 18mpbird 156 . . 3 ((𝐹𝐺 (Fun 𝐹 A dom 𝐹)) → (𝐹A) (𝐺 “ {A}))
2019exp32 347 . 2 (𝐹𝐺 → (Fun 𝐹 → (A dom 𝐹 → (𝐹A) (𝐺 “ {A}))))
2120impcom 116 1 ((Fun 𝐹 𝐹𝐺) → (A dom 𝐹 → (𝐹A) (𝐺 “ {A})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  {csn 3367  cop 3370  dom cdm 4288  cima 4291  Fun wfun 4839  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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