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Theorem elrnmpt1s 5828
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt1s.1 (𝑥 = 𝐷𝐵 = 𝐶)
Assertion
Ref Expression
elrnmpt1s ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐴   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmpt1s
StepHypRef Expression
1 eqid 2826 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . 4 (𝑥 = 𝐷𝐵 = 𝐶)
32rspceeqv 3642 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
41, 3mpan2 687 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
5 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5827 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
76biimparc 480 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
84, 7sylan 580 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wrex 3144  cmpt 5143  ran crn 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-mpt 5144  df-cnv 5562  df-dm 5564  df-rn 5565
This theorem is referenced by:  wunex2  10154  dfod2  18627  dprd2dlem1  19099  dprd2da  19100  ordtbaslem  21731  subgntr  22649  opnsubg  22650  tgpconncomp  22655  tsmsxplem1  22695  xrge0gsumle  23375  xrge0tsms  23376  minveclem3b  23965  minveclem3  23966  minveclem4  23969  efsubm  25067  dchrisum0fno1  26020  fnpreimac  30350  xrge0tsmsd  30625  esumcvg  31250  esum2d  31257  msubco  32681  suprubrnmpt2  41408  infxrlbrnmpt2  41568  sge0xaddlem1  42600
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