Theorem elrnmpt1d 5919

Description: Elementhood in an image set. Deducion version of elrnmpt1 5915. (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 5915	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
5	1, 2, 4	syl2anc 585	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ↦ 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-sbc 3729  df-csb 3838  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:  mptcnfimad  7939  elrgspnsubrunlem1  33308  rnmptbd2lem  45677  rnmptbdlem  45684  rnmptss2  45686  rnmptssbi  45689  supminfxrrnmpt  45899  sge0f1o  46810