Theorem elrnmptd 5918

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 5913	. . 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 1542   ∈ wcel 2114  ∃wrex 3061   ↦ cmpt 5166  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  pwfilem  9228  evls1maprnss  22343  elrgspnlem1  33303  elrgspnlem2  33304  elrgspnlem3  33305  nsgmgc  33472  nsgqusf1olem1  33473  algextdeglem4  33864  zarclsun  34014  rnmptssrn  45612  infnsuprnmpt  45679  supminfrnmpt  45873  supminfxrrnmpt  45899  sge0sup  46819  sge0resplit  46834  sge0xaddlem2  46862  sge0pnfmpt  46873  sge0reuz  46875  sge0reuzb  46876  hoidmvlelem2  47024