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Theorem elrnmptd 40120
Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elrnmptd.f 𝐹 = (𝑥𝐴𝐵)
elrnmptd.x (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
elrnmptd.c (𝜑𝐶𝑉)
Assertion
Ref Expression
elrnmptd (𝜑𝐶 ∈ ran 𝐹)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptd
StepHypRef Expression
1 elrnmptd.x . 2 (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
2 elrnmptd.c . . 3 (𝜑𝐶𝑉)
3 elrnmptd.f . . . 4 𝐹 = (𝑥𝐴𝐵)
43elrnmpt 5576 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
61, 5mpbird 249 1 (𝜑𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = 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:  infnsuprnmpt  40212  supminfrnmpt  40415  supminfxrrnmpt  40444  sge0sup  41351  sge0resplit  41366  sge0xaddlem2  41394  sge0pnfmpt  41405  sge0reuz  41407  sge0reuzb  41408  hoidmvlelem2  41556
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