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Theorem elrnmptd 5798
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 5793 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
61, 5mpbird 260 1 (𝜑𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wcel 2113  wrex 3054  cmpt 5107  ran crn 5520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-rex 3059  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-mpt 5108  df-cnv 5527  df-dm 5529  df-rn 5530
This theorem is referenced by:  pwfilem  8768  nsgmgc  31161  nsgqusf1olem1  31162  zarclsun  31384  rnmptssrn  42241  infnsuprnmpt  42317  supminfrnmpt  42507  supminfxrrnmpt  42535  sge0sup  43455  sge0resplit  43470  sge0xaddlem2  43498  sge0pnfmpt  43509  sge0reuz  43511  sge0reuzb  43512  hoidmvlelem2  43660
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