Theorem elmptima 42693

Description: The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Assertion

Ref	Expression
elmptima	⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem elmptima

Step	Hyp	Ref	Expression
1		mptima 5970	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)
2	1	a1i 11	. . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵))
3	2	eleq2d 2824	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)))
4		eqid 2738	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)
5	4	elrnmpt 5854	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))
6	3, 5	bitrd 278	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ∩ cin 3882   ↦ cmpt 5153  ran crn 5581   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  liminfvalxr  43214