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Theorem fvelrn 5598
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)

Proof of Theorem fvelrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2220 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 460 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 fveq2 5468 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eleq1d 2226 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ ran 𝐹 ↔ (𝐹𝐴) ∈ ran 𝐹))
52, 4imbi12d 233 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)))
6 funfvop 5579 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
7 vex 2715 . . . . . 6 𝑥 ∈ V
8 opeq1 3741 . . . . . . 7 (𝑦 = 𝑥 → ⟨𝑦, (𝐹𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩)
98eleq1d 2226 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑦, (𝐹𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
107, 9spcev 2807 . . . . 5 (⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹 → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
116, 10syl 14 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
12 funfvex 5485 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
13 elrn2g 4776 . . . . 5 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹))
1412, 13syl 14 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹))
1511, 14mpbird 166 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
165, 15vtoclg 2772 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹))
1716anabsi7 571 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wex 1472  wcel 2128  Vcvv 2712  cop 3563  dom cdm 4586  ran crn 4587  Fun wfun 5164  cfv 5170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by:  fnfvelrn  5599  eldmrexrn  5608  funfvima  5698  elunirn  5716  frecuzrdgdomlem  10316  frecuzrdgsuctlem  10322
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