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Theorem rnmptc 6945
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 5590 . . . . 5 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
42, 3eqtr4i 2846 . . . 4 𝐹 = (𝐴 × {𝐵})
54rneqi 5783 . . 3 ran 𝐹 = ran (𝐴 × {𝐵})
6 rnxp 6003 . . 3 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
75, 6syl5eq 2867 . 2 (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
81, 7syl 17 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wne 3006  c0 4269  {csn 4543  cmpt 5122   × cxp 5529  ran crn 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-mpt 5123  df-xp 5537  df-rel 5538  df-cnv 5539  df-dm 5541  df-rn 5542
This theorem is referenced by:  qsalrel  39237  limsup0  42127  limsuppnfdlem  42134  limsup10ex  42206  liminf10ex  42207  fourierdlem60  42599  fourierdlem61  42600  sge0z  42805
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