Theorem elmptima 46014

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 6079	. . . 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 2852	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)))
4		eqid 2766	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐵)
5	4	elrnmpt 5953	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ∩ cin 3907   ↦ cmpt 5197  ran crn 5667   “ cima 5669

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-mpt 5198  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by:  liminfvalxr  46538