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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 2731 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . 4 (𝑥 = 𝐷𝐵 = 𝐶)
32rspceeqv 3633 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
41, 3mpan2 688 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
5 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5955 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
76biimparc 479 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
84, 7sylan 579 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wrex 3069  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 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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  10737  dfod2  19474  dprd2dlem1  19953  dprd2da  19954  ordtbaslem  22913  subgntr  23832  opnsubg  23833  tgpconncomp  23838  tsmsxplem1  23878  xrge0gsumle  24570  xrge0tsms  24571  minveclem3b  25177  minveclem3  25178  minveclem4  25181  efsubm  26297  dchrisum0fno1  27251  fnpreimac  32164  xrge0tsmsd  32480  esumcvg  33383  esum2d  33390  msubco  34821  suprubrnmpt2  44255  infxrlbrnmpt2  44419  sge0xaddlem1  45448
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