Theorem elrnmptd 5798

Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
elrnmptd.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmptd.x	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
elrnmptd.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
elrnmptd	⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptd

Step	Hyp	Ref	Expression
1		elrnmptd.x	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
2		elrnmptd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
3		elrnmptd.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt 5793	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
5	2, 4	syl 17	. 2 ⊢ (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
6	1, 5	mpbird 260	1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1542   ∈ wcel 2113  ∃wrex 3054   ↦ cmpt 5107  ran crn 5520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-rex 3059  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-mpt 5108  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by:  pwfilem  8768  nsgmgc  31161  nsgqusf1olem1  31162  zarclsun  31384  rnmptssrn  42241  infnsuprnmpt  42317  supminfrnmpt  42507  supminfxrrnmpt  42535  sge0sup  43455  sge0resplit  43470  sge0xaddlem2  43498  sge0pnfmpt  43509  sge0reuz  43511  sge0reuzb  43512  hoidmvlelem2  43660