Theorem elrnmptdv 5953

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 3584	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)
4		elrnmptdv.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		elrnmptdv.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	elrnmpt 5946	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ↦ cmpt 5190  ran crn 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is used by:  cycsubggend  19339  nsgqusf1olem3  33852  zart0  34397  rr-elrnmpt3d  45054