Theorem elrnmptd 41447

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 5830	. . 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 259	1 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ↦ cmpt 5148  ran crn 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-mpt 5149  df-cnv 5565  df-dm 5567  df-rn 5568

This theorem is referenced by:  rnmptssrn  41449  infnsuprnmpt  41529  supminfrnmpt  41726  supminfxrrnmpt  41754  sge0sup  42680  sge0resplit  42695  sge0xaddlem2  42723  sge0pnfmpt  42734  sge0reuz  42736  sge0reuzb  42737  hoidmvlelem2  42885