Theorem rr-elrnmpt3d 44858

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 2775	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)
6	1, 2, 3, 5	elrnmptdv 5956	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5196  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 5261  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-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  mnurndlem1  44917