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Theorem rnmptc 7219
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 5740 . . . . 5 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
42, 3eqtr4i 2759 . . . 4 𝐹 = (𝐴 × {𝐵})
54rneqi 5939 . . 3 ran 𝐹 = ran (𝐴 × {𝐵})
6 rnxp 6174 . . 3 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
75, 6eqtrid 2780 . 2 (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
81, 7syl 17 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wne 2937  c0 4323  {csn 4629  cmpt 5231   × cxp 5676  ran crn 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  mptiffisupp  32486  qsalrel  41731  limsup0  45082  limsuppnfdlem  45089  limsup10ex  45161  liminf10ex  45162  fourierdlem60  45554  fourierdlem61  45555  sge0z  45763
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