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Theorem elrnmpt1s 5956
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 2732 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . 4 (𝑥 = 𝐷𝐵 = 𝐶)
32rspceeqv 3633 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
41, 3mpan2 689 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
5 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5955 . . 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 1541  wcel 2106  wrex 3070  cmpt 5231  ran crn 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687
This theorem is referenced by:  wunex2  10735  dfod2  19434  dprd2dlem1  19913  dprd2da  19914  ordtbaslem  22699  subgntr  23618  opnsubg  23619  tgpconncomp  23624  tsmsxplem1  23664  xrge0gsumle  24356  xrge0tsms  24357  minveclem3b  24952  minveclem3  24953  minveclem4  24956  efsubm  26067  dchrisum0fno1  27021  fnpreimac  31934  xrge0tsmsd  32250  esumcvg  33153  esum2d  33160  msubco  34591  suprubrnmpt2  44035  infxrlbrnmpt2  44199  sge0xaddlem1  45228
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