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Theorem elrnmpt1d 5974
Description: Elementhood in an image set. Deducion version of elrnmpt1 5970. (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 5970 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
51, 2, 4syl2anc 584 1 (𝜑𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cmpt 5224  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  mptcnfimad  8012  elrgspnsubrunlem1  33252  rnmptbd2lem  45260  rnmptbdlem  45267  rnmptss2  45269  rnmptssbi  45272  supminfxrrnmpt  45487  sge0f1o  46402
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