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Theorem rnmptc 7157
Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
Hypotheses
Ref Expression
rnmptc.f 𝐹 = (𝑥𝐴𝐵)
rnmptc.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptc (𝜑 → ran 𝐹 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptc
StepHypRef Expression
1 rnmptc.a . 2 (𝜑𝐴 ≠ ∅)
2 rnmptc.f . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 fconstmpt 5695 . . . . 5 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
42, 3eqtr4i 2764 . . . 4 𝐹 = (𝐴 × {𝐵})
54rneqi 5893 . . 3 ran 𝐹 = ran (𝐴 × {𝐵})
6 rnxp 6123 . . 3 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
75, 6eqtrid 2785 . 2 (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
81, 7syl 17 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wne 2940  c0 4283  {csn 4587  cmpt 5189   × cxp 5632  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-mpt 5190  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  mptiffisupp  31654  qsalrel  40710  limsup0  44021  limsuppnfdlem  44028  limsup10ex  44100  liminf10ex  44101  fourierdlem60  44493  fourierdlem61  44494  sge0z  44702
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