Theorem elrnmptd 5973

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 5968	. . 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 257	1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∃wrex 3069   ↦ cmpt 5224  ran crn 5685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-cnv 5692  df-dm 5694  df-rn 5695

This theorem is referenced by:  pwfilem  9357  evls1maprnss  22383  elrgspnlem1  33247  elrgspnlem2  33248  elrgspnlem3  33249  nsgmgc  33441  nsgqusf1olem1  33442  algextdeglem4  33762  zarclsun  33870  rnmptssrn  45192  infnsuprnmpt  45262  supminfrnmpt  45461  supminfxrrnmpt  45487  sge0sup  46411  sge0resplit  46426  sge0xaddlem2  46454  sge0pnfmpt  46465  sge0reuz  46467  sge0reuzb  46468  hoidmvlelem2  46616