Theorem elrnmptdv 5860

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
elrnmptdv.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmptdv.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)
elrnmptdv.3	⊢ (𝜑 → 𝐷 ∈ 𝑉)
elrnmptdv.4	⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)

Assertion

Ref	Expression
elrnmptdv	⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Distinct variable groups:   𝑥,𝐷   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptdv

Step	Hyp	Ref	Expression
1		elrnmptdv.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)
2		elrnmptdv.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3	1, 2	rspcime 3556	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)
4		elrnmptdv.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		elrnmptdv.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	elrnmpt 5854	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
7	4, 6	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
8	3, 7	mpbird 256	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ↦ cmpt 5153  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  cycsubggend  18739  nsgqusf1olem3  31502  zart0  31731  rr-elrnmpt3d  41708