Theorem elrnmptdv 4978

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 2914	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐷 = 𝐵)
4		elrnmptdv.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		elrnmptdv.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	elrnmpt 4973	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
7	4, 6	syl 14	. 2 ⊢ (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐷 = 𝐵))
8	3, 7	mpbird 167	1 ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ↦ cmpt 4145  ran crn 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-mpt 4147  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by: (None)