Theorem rnmptc 7205

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 5723	. . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	eqtr4i 2789	. . . 4 ⊢ 𝐹 = (𝐴 × {𝐵})
5	4	rneqi 5927	. . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵})
6		rnxp 6168	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	eqtrid 2810	. 2 ⊢ (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
8	1, 7	syl 18	1 ⊢ (𝜑 → ran 𝐹 = {𝐵})

Syntax hints:   → wi 4   = wceq 1570   ≠ wne 2958  ∅c0 4286  {csn 4589   ↦ cmpt 5192   × cxp 5659  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  mptiffisupp  33038  qsalrel  43009  limsup0  46408  limsuppnfdlem  46415  limsup10ex  46487  liminf10ex  46488  fourierdlem60  46880  fourierdlem61  46881  sge0z  47089