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Theorem elrnmpt1d 5975
Description: Elementhood in an image set. Deducion version of elrnmpt1 5971. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elrnmpt1d.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt1d.2 (𝜑𝑥𝐴)
elrnmpt1d.3 (𝜑𝐵𝑉)
Assertion
Ref Expression
elrnmpt1d (𝜑𝐵 ∈ ran 𝐹)

Proof of Theorem elrnmpt1d
StepHypRef Expression
1 elrnmpt1d.2 . 2 (𝜑𝑥𝐴)
2 elrnmpt1d.3 . 2 (𝜑𝐵𝑉)
3 elrnmpt1d.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmpt1 5971 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
51, 2, 4syl2anc 584 1 (𝜑𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  mptcnfimad  8011  elrgspnsubrunlem1  33251  rnmptbd2lem  45255  rnmptbdlem  45262  rnmptss2  45264  rnmptssbi  45267  supminfxrrnmpt  45482  sge0f1o  46397
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