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Theorem elrnmpt1d 5978
Description: Elementhood in an image set. Deducion version of elrnmpt1 5974. (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 5974 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
51, 2, 4syl2anc 584 1 (𝜑𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  mptcnfimad  8010  rnmptbd2lem  45193  rnmptbdlem  45200  rnmptss2  45202  rnmptssbi  45206  supminfxrrnmpt  45421  sge0f1o  46338
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