Theorem rr-elrnmpt3d 44665

Description: Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
rr-elrnmpt3d.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rr-elrnmpt3d.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)
rr-elrnmpt3d.3	⊢ (𝜑 → 𝐷 ∈ 𝑉)
rr-elrnmpt3d.4	⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷)

Assertion

Ref	Expression
rr-elrnmpt3d	⊢ (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 rr-elrnmpt3d

Step	Hyp	Ref	Expression
1		rr-elrnmpt3d.1	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		rr-elrnmpt3d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		rr-elrnmpt3d.3	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
4		rr-elrnmpt3d.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷)
5	4	eqcomd 2747	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)
6	1, 2, 3, 5	elrnmptdv 5913	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ↦ cmpt 5155  ran crn 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-mpt 5156  df-cnv 5628  df-dm 5630  df-rn 5631

This theorem is referenced by:  mnurndlem1  44738