MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funfvima3 Structured version   Visualization version   GIF version

Theorem funfvima3 7191
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 3942 . . . . 5 (𝐹𝐺 → (⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹 → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
2 funfvop 7005 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
31, 2impel 507 . . . 4 ((𝐹𝐺 ∧ (Fun 𝐹𝐴 ∈ dom 𝐹)) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺)
4 sneq 4601 . . . . . . . 8 (𝑥 = 𝐴 → {𝑥} = {𝐴})
54imaeq2d 6018 . . . . . . 7 (𝑥 = 𝐴 → (𝐺 “ {𝑥}) = (𝐺 “ {𝐴}))
65eleq2d 2824 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
7 opeq1 4835 . . . . . . 7 (𝑥 = 𝐴 → ⟨𝑥, (𝐹𝐴)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
87eleq1d 2823 . . . . . 6 (𝑥 = 𝐴 → (⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
9 vex 3452 . . . . . . 7 𝑥 ∈ V
10 fvex 6860 . . . . . . 7 (𝐹𝐴) ∈ V
119, 10elimasn 6046 . . . . . 6 ((𝐹𝐴) ∈ (𝐺 “ {𝑥}) ↔ ⟨𝑥, (𝐹𝐴)⟩ ∈ 𝐺)
126, 8, 11vtoclbg 3531 . . . . 5 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ (𝐺 “ {𝐴}) ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐺))
1312ad2antll 728 . . . 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 422 . 2 (𝐹𝐺 → (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴}))))
1615impcom 409 1 ((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wss 3915  {csn 4591  cop 4597  dom cdm 5638  cima 5641  Fun wfun 6495  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  dfac3  10064
  Copyright terms: Public domain W3C validator