Theorem elmptima 41896

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 5908	. . . 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 2875	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)))
4		eqid 2798	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)
5	4	elrnmpt 5792	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))
6	3, 5	bitrd 282	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ∩ cin 3880   ↦ cmpt 5110  ran crn 5520   “ cima 5522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532

This theorem is referenced by:  liminfvalxr  42425