Theorem rr-elrnmpt3d 45349

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 2769	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵)
6	1, 2, 3, 5	elrnmptdv 5952	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ↦ cmpt 5189  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  mnurndlem1  45408