Theorem elrnmptdv 5955

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 3585	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)
4		elrnmptdv.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		elrnmptdv.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	elrnmpt 5948	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
7	4, 6	syl 18	. 2 ⊢ (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
8	3, 7	mpbird 260	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087   ↦ cmpt 5191  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-mpt 5192  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  cycsubggend  19275  nsgqusf1olem3  33690  zart0  34235  rr-elrnmpt3d  44902