Theorem elrnmpt1d 44230

Description: Elementhood in an image set. (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 5956	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
5	1, 2, 4	syl2anc 582	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5230  ran crn 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5683  df-dm 5685  df-rn 5686

This theorem is referenced by:  rnmptbd2lem  44250  rnmptbdlem  44257  rnmptss2  44259  rnmptssbi  44263  supminfxrrnmpt  44479