Theorem opnneilv 48919

Description: The converse of opnneir 48917 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 48915), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)

Hypotheses

Ref	Expression
opnneir.1	⊢ (𝜑 → 𝐽 ∈ Top)
opnneilv.2	⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))

Assertion

Ref	Expression
opnneilv	⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Distinct variable groups:   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem opnneilv

Step	Hyp	Ref	Expression
1		df-rex 3055	. 2 ⊢ (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓))
2		opnneir.1	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
3		neii2 23016	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
4	2, 3	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
5	4	r19.41dv 48812	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓))
6	5	expl 457	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓)))
7		anass 468	. . . . . 6 ⊢ (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)))
8		opnneilv.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))
9	8	expimpd 453	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝜒))
10	9	anim2d 612	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
11	7, 10	biimtrid 242	. . . . 5 ⊢ (𝜑 → (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
12	11	reximdv 3145	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
13	6, 12	syld 47	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
14	13	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
15	1, 14	biimtrid 242	1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780   ∈ wcel 2110  ∃wrex 3054   ⊆ wss 3900  ‘cfv 6477  Topctop 22801  neicnei 23005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-top 22802  df-nei 23006

This theorem is referenced by:  opnneil  48920