Theorem elrnmpt1d 5955

Description: Elementhood in an image set. Deducion version of elrnmpt1 5951. (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 5951	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
5	1, 2, 4	syl2anc 595	1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5194  ran crn 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  mptcnfimad  7983  elrgspnsubrunlem1  33508  rnmptbd2lem  45890  rnmptbdlem  45897  rnmptss2  45899  rnmptssbi  45902  supminfxrrnmpt  46112  sge0f1o  47023