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Theorem funfvima3 5313
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 5200 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐹)
2 ssel 2912 . . . . . 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 ax-ia2 100 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → A dom 𝐹)
6 sneq 3357 . . . . . . . . . 10 (x = A → {x} = {A})
76imaeq2d 4591 . . . . . . . . 9 (x = A → (𝐺 “ {x}) = (𝐺 “ {A}))
87eleq2d 2085 . . . . . . . 8 (x = A → ((𝐹A) (𝐺 “ {x}) ↔ (𝐹A) (𝐺 “ {A})))
9 opeq1 3519 . . . . . . . . 9 (x = A → ⟨x, (𝐹A)⟩ = ⟨A, (𝐹A)⟩)
109eleq1d 2084 . . . . . . . 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 2534 . . . . . . 7 x V
14 funfvex 5113 . . . . . . 7 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
15 elimasng 4616 . . . . . . 7 ((x V (𝐹A) V) → ((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺))
1613, 14, 15sylancr 395 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) (𝐺 “ {x}) ↔ ⟨x, (𝐹A)⟩ 𝐺))
175, 12, 16vtocld 2579 . . . . 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 1226   wcel 1370  Vcvv 2531  wss 2890  {csn 3346  cop 3349  dom cdm 4268  cima 4271  Fun wfun 4819  cfv 4825
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by: (None)
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