Theorem rnmptc 7163

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 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptc.a	⊢ (𝜑 → 𝐴 ≠ ∅)

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

Step	Hyp	Ref	Expression
1		rnmptc.a	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		rnmptc.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		fconstmpt 5694	. . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	eqtr4i 2763	. . . 4 ⊢ 𝐹 = (𝐴 × {𝐵})
5	4	rneqi 5894	. . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵})
6		rnxp 6136	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	eqtrid 2784	. 2 ⊢ (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
8	1, 7	syl 17	1 ⊢ (𝜑 → ran 𝐹 = {𝐵})

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933  ∅c0 4287  {csn 4582   ↦ cmpt 5181   × cxp 5630  ran crn 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643

This theorem is referenced by:  mptiffisupp  32783  qsalrel  42611  limsup0  46052  limsuppnfdlem  46059  limsup10ex  46131  liminf10ex  46132  fourierdlem60  46524  fourierdlem61  46525  sge0z  46733