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Theorem rnmptc 40277
Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
rnmptc.f 𝐹 = (𝑥𝐴𝐵)
rnmptc.b ((𝜑𝑥𝐴) → 𝐵𝐶)
rnmptc.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptc (𝜑 → ran 𝐹 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem rnmptc
StepHypRef Expression
1 rnmptc.f . . 3 𝐹 = (𝑥𝐴𝐵)
2 fconstmpt 5411 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2eqtr4i 2805 . 2 𝐹 = (𝐴 × {𝐵})
4 rnmptc.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝐶)
54, 1fmptd 6648 . . . 4 (𝜑𝐹:𝐴𝐶)
65ffnd 6292 . . 3 (𝜑𝐹 Fn 𝐴)
7 rnmptc.a . . 3 (𝜑𝐴 ≠ ∅)
8 fconst5 6743 . . 3 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
96, 7, 8syl2anc 579 . 2 (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
103, 9mpbii 225 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wne 2969  c0 4141  {csn 4398  cmpt 4965   × cxp 5353  ran crn 5356   Fn wfn 6130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143
This theorem is referenced by:  limsup0  40834  limsuppnfdlem  40841  limsup10ex  40913  liminf10ex  40914  fourierdlem60  41310  fourierdlem61  41311  sge0z  41516
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