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 5693	. . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	eqtr4i 2755	. . . 4 ⊢ 𝐹 = (𝐴 × {𝐵})
5	4	rneqi 5890	. . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵})
6		rnxp 6131	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	eqtrid 2776	. 2 ⊢ (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
8	1, 7	syl 17	1 ⊢ (𝜑 → ran 𝐹 = {𝐵})

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2925  ∅c0 4292  {csn 4585   ↦ cmpt 5183   × cxp 5629  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  mptiffisupp  32589  qsalrel  42201  limsup0  45665  limsuppnfdlem  45672  limsup10ex  45744  liminf10ex  45745  fourierdlem60  46137  fourierdlem61  46138  sge0z  46346