Theorem rnmptc 7207

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 5717	. . . . 5 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	2, 3	eqtr4i 2786	. . . 4 ⊢ 𝐹 = (𝐴 × {𝐵})
5	4	rneqi 5921	. . 3 ⊢ ran 𝐹 = ran (𝐴 × {𝐵})
6		rnxp 6163	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	eqtrid 2807	. 2 ⊢ (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
8	1, 7	syl 18	1 ⊢ (𝜑 → ran 𝐹 = {𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2955  ∅c0 4279  {csn 4584   ↦ cmpt 5186   × cxp 5653  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  mptiffisupp  33166  qsalrel  43109  limsup0  46523  limsuppnfdlem  46530  limsup10ex  46602  liminf10ex  46603  fourierdlem60  46995  fourierdlem61  46996  sge0z  47204