Theorem rr-elrnmpt3d 44221

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 2743	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)
6	1, 2, 3, 5	elrnmptdv 5976	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5225  ran crn 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  mnurndlem1  44300