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Theorem elrnmpt1s 5822
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 2818 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . 4 (𝑥 = 𝐷𝐵 = 𝐶)
32rspceeqv 3635 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
41, 3mpan2 687 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
5 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5821 . . 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 1528  wcel 2105  wrex 3136  cmpt 5137  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  wunex2  10148  dfod2  18620  dprd2dlem1  19092  dprd2da  19093  ordtbaslem  21724  subgntr  22642  opnsubg  22643  tgpconncomp  22648  tsmsxplem1  22688  xrge0gsumle  23368  xrge0tsms  23369  minveclem3b  23958  minveclem3  23959  minveclem4  23962  efsubm  25062  dchrisum0fno1  26014  fnpreimac  30344  xrge0tsmsd  30619  esumcvg  31244  esum2d  31251  msubco  32675  suprubrnmpt2  41400  infxrlbrnmpt2  41560  sge0xaddlem1  42592
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