Theorem elrnmpt1d 5942

Description: Elementhood in an image set. Deducion version of elrnmpt1 5938. (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 5938	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
5	1, 2, 4	syl2anc 593	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ↦ cmpt 5183  ran crn 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-mpt 5184  df-cnv 5657  df-dm 5659  df-rn 5660

This theorem is referenced by:  mptcnfimad  7969  elrgspnsubrunlem1  33430  rnmptbd2lem  45828  rnmptbdlem  45835  rnmptss2  45837  rnmptssbi  45840  supminfxrrnmpt  46050  sge0f1o  46961