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Theorem elrnmpt1s 5577
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 2799 . . 3 𝐶 = 𝐶
2 elrnmpt1s.1 . . . 4 (𝑥 = 𝐷𝐵 = 𝐶)
32rspceeqv 3515 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
41, 3mpan2 683 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
5 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5576 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
76biimparc 472 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
84, 7sylan 576 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wrex 3090  cmpt 4922  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-mpt 4923  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  wunex2  9848  dfod2  18294  dprd2dlem1  18756  dprd2da  18757  ordtbaslem  21321  subgntr  22238  opnsubg  22239  tgpconncomp  22244  tsmsxplem1  22284  xrge0gsumle  22964  xrge0tsms  22965  minveclem3b  23538  minveclem3  23539  minveclem4  23542  efsubm  24639  dchrisum0fno1  25552  xrge0tsmsd  30301  esumcvg  30664  esum2d  30671  msubco  31945  suprubrnmpt2  40214  infxrlbrnmpt2  40380  sge0xaddlem1  41393
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