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Theorem elrnmptd 39883
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 5527 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
61, 5mpbird 247 1 (𝜑𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wrex 3051  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  infnsuprnmpt  39982  supminfrnmpt  40188  supminfxrrnmpt  40217  sge0sup  41129  sge0resplit  41144  sge0xaddlem2  41172  sge0pnfmpt  41183  sge0reuz  41185  sge0reuzb  41186  hoidmvlelem2  41334
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