Theorem elrnmpt1d 5974

Description: Elementhood in an image set. Deducion version of elrnmpt1 5970. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
elrnmpt1d.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt1d.2	⊢ (𝜑 → 𝑥 ∈ 𝐴)
elrnmpt1d.3	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
elrnmpt1d	⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Proof of Theorem elrnmpt1d

Step	Hyp	Ref	Expression
1		elrnmpt1d.2	. 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
2		elrnmpt1d.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3		elrnmpt1d.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt1 5970	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
5	1, 2, 4	syl2anc 584	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ↦ 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-sbc 3788  df-csb 3899  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:  mptcnfimad  8012  elrgspnsubrunlem1  33252  rnmptbd2lem  45260  rnmptbdlem  45267  rnmptss2  45269  rnmptssbi  45272  supminfxrrnmpt  45487  sge0f1o  46402