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Theorem elrnmpt1s 5343
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 2621 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
32eqeq2d 2631 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
43rspcev 3299 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4mpan2 706 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
6 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
76elrnmpt 5342 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
87biimparc 504 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
95, 8sylan 488 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  cmpt 4683  ran crn 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-mpt 4685  df-cnv 5092  df-dm 5094  df-rn 5095
This theorem is referenced by:  wunex2  9520  dfod2  17921  dprd2dlem1  18380  dprd2da  18381  ordtbaslem  20932  subgntr  21850  opnsubg  21851  tgpconncomp  21856  tsmsxplem1  21896  xrge0gsumle  22576  xrge0tsms  22577  minveclem3b  23139  minveclem3  23140  minveclem4  23143  efsubm  24235  dchrisum0fno1  25134  xrge0tsmsd  29612  esumcvg  29971  esum2d  29978  msubco  31189  suprubrnmpt2  38978  infxrlbrnmpt2  39136  sge0xaddlem1  39987
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