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Theorem rnmptc 7203
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 5731 . . . . 5 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
42, 3eqtr4i 2757 . . . 4 𝐹 = (𝐴 × {𝐵})
54rneqi 5929 . . 3 ran 𝐹 = ran (𝐴 × {𝐵})
6 rnxp 6162 . . 3 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
75, 6eqtrid 2778 . 2 (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
81, 7syl 17 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wne 2934  c0 4317  {csn 4623  cmpt 5224   × cxp 5667  ran crn 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  mptiffisupp  32420  qsalrel  41604  limsup0  44963  limsuppnfdlem  44970  limsup10ex  45042  liminf10ex  45043  fourierdlem60  45435  fourierdlem61  45436  sge0z  45644
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