Theorem elmptima 42804

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 5981	. . . 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 5865	. 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 2106  ∃wrex 3065   ∩ cin 3886   ↦ cmpt 5157  ran crn 5590   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  liminfvalxr  43324