Theorem elrnmptd 5959

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 5954	. . 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 256	1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  ∃wrex 3068   ↦ cmpt 5230  ran crn 5676

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5683  df-dm 5685  df-rn 5686

This theorem is referenced by:  pwfilem  9179  nsgmgc  32797  nsgqusf1olem1  32798  evls1maprnss  33050  algextdeglem4  33065  zarclsun  33148  rnmptssrn  44179  infnsuprnmpt  44252  supminfrnmpt  44453  supminfxrrnmpt  44479  sge0sup  45405  sge0resplit  45420  sge0xaddlem2  45448  sge0pnfmpt  45459  sge0reuz  45461  sge0reuzb  45462  hoidmvlelem2  45610