Theorem suplocexprlemmu 7716

Description: Lemma for suplocexpr 7723. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemmu	⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))

Distinct variable groups:   𝐴,𝑠,𝑢,𝑤   𝑥,𝐴,𝑦,𝑠,𝑢   𝐵,𝑠   𝜑,𝑠,𝑢,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐴(𝑧)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑢)

Proof of Theorem suplocexprlemmu

Dummy variables 𝑗 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.ub	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
2		prop 7473	. . . . . . 7 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
3		prmu 7476	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
4	2, 3	syl 14	. . . . . 6 ⊢ (𝑥 ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
5	4	ad2antrl 490	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
6		fo2nd 6158	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
7		fofun 5439	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
9		fvelima 5567	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ 𝑡 ∈ (2nd “ 𝐴)) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
10	8, 9	mpan 424	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd “ 𝐴) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
11	10	adantl 277	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
12		suplocexpr.m	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
13		suplocexpr.loc	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
14	12, 1, 13	suplocexprlemss 7713	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ P)
15	14	ad5antr 496	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝐴 ⊆ P)
16		simprl 529	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢 ∈ 𝐴)
17	15, 16	sseldd 3156	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢 ∈ P)
18		simprl 529	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → 𝑥 ∈ P)
19	18	ad4antr 494	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑥 ∈ P)
20		breq1 4006	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝑦<P 𝑥 ↔ 𝑢<P 𝑥))
21		simprr 531	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
22	21	ad4antr 494	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
23	20, 22, 16	rspcdva 2846	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢<P 𝑥)
24		ltsopr 7594	. . . . . . . . . . . . . . . . 17 ⊢ <P Or P
25		so2nr 4321	. . . . . . . . . . . . . . . . 17 ⊢ ((<P Or P ∧ (𝑢 ∈ P ∧ 𝑥 ∈ P)) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
26	24, 25	mpan 424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ P ∧ 𝑥 ∈ P) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
27	17, 19, 26	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
28		imnan 690	. . . . . . . . . . . . . . 15 ⊢ ((𝑢<P 𝑥 → ¬ 𝑥<P 𝑢) ↔ ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
29	27, 28	sylibr 134	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (𝑢<P 𝑥 → ¬ 𝑥<P 𝑢))
30	23, 29	mpd 13	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ¬ 𝑥<P 𝑢)
31		aptiprlemu 7638	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ P ∧ 𝑥 ∈ P ∧ ¬ 𝑥<P 𝑢) → (2nd ‘𝑥) ⊆ (2nd ‘𝑢))
32	17, 19, 30, 31	syl3anc 1238	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (2nd ‘𝑥) ⊆ (2nd ‘𝑢))
33		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ (2nd ‘𝑥))
34	32, 33	sseldd 3156	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ (2nd ‘𝑢))
35		simprr 531	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (2nd ‘𝑢) = 𝑡)
36	34, 35	eleqtrd 2256	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ 𝑡)
37	11, 36	rexlimddv 2599	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) → 𝑠 ∈ 𝑡)
38	37	ralrimiva 2550	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) → ∀𝑡 ∈ (2nd “ 𝐴)𝑠 ∈ 𝑡)
39		vex 2740	. . . . . . . . 9 ⊢ 𝑠 ∈ V
40	39	elint2 3851	. . . . . . . 8 ⊢ (𝑠 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑠 ∈ 𝑡)
41	38, 40	sylibr 134	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) → 𝑠 ∈ ∩ (2nd “ 𝐴))
42	41	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) → (𝑠 ∈ (2nd ‘𝑥) → 𝑠 ∈ ∩ (2nd “ 𝐴)))
43	42	reximdva 2579	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → (∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥) → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴)))
44	5, 43	mpd 13	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴))
45	1, 44	rexlimddv 2599	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴))
46		simprr 531	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 ∈ ∩ (2nd “ 𝐴))
47		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 ∈ Q)
48		1nq 7364	. . . . . . . . 9 ⊢ 1Q ∈ Q
49		addclnq 7373	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 1Q ∈ Q) → (𝑠 +Q 1Q) ∈ Q)
50	47, 48, 49	sylancl 413	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51		ltaddnq 7405	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 1Q ∈ Q) → 𝑠 <Q (𝑠 +Q 1Q))
52	47, 48, 51	sylancl 413	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53		breq2 4007	. . . . . . . . 9 ⊢ (𝑗 = (𝑠 +Q 1Q) → (𝑠 <Q 𝑗 ↔ 𝑠 <Q (𝑠 +Q 1Q)))
54	53	rspcev 2841	. . . . . . . 8 ⊢ (((𝑠 +Q 1Q) ∈ Q ∧ 𝑠 <Q (𝑠 +Q 1Q)) → ∃𝑗 ∈ Q 𝑠 <Q 𝑗)
55	50, 52, 54	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q 𝑠 <Q 𝑗)
56		breq1 4006	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (𝑤 <Q 𝑗 ↔ 𝑠 <Q 𝑗))
57	56	rexbidv 2478	. . . . . . . 8 ⊢ (𝑤 = 𝑠 → (∃𝑗 ∈ Q 𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q 𝑠 <Q 𝑗))
58	57	rspcev 2841	. . . . . . 7 ⊢ ((𝑠 ∈ ∩ (2nd “ 𝐴) ∧ ∃𝑗 ∈ Q 𝑠 <Q 𝑗) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗)
59	46, 55, 58	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗)
60		rexcom 2641	. . . . . 6 ⊢ (∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)
61	59, 60	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)
62		ssid 3175	. . . . . 6 ⊢ Q ⊆ Q
63		rexss 3222	. . . . . 6 ⊢ (Q ⊆ Q → (∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
64	62, 63	ax-mp 5	. . . . 5 ⊢ (∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
65	61, 64	sylib 122	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
66		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
67	66	suplocexprlem2b 7712	. . . . . . . . 9 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
68	14, 67	syl 14	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
69	68	eleq2d 2247	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ (2nd ‘𝐵) ↔ 𝑗 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
70		breq2 4007	. . . . . . . . 9 ⊢ (𝑢 = 𝑗 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑗))
71	70	rexbidv 2478	. . . . . . . 8 ⊢ (𝑢 = 𝑗 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
72	71	elrab 2893	. . . . . . 7 ⊢ (𝑗 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
73	69, 72	bitrdi 196	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (2nd ‘𝐵) ↔ (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
74	73	rexbidv 2478	. . . . 5 ⊢ (𝜑 → (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
75	74	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
76	65, 75	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵))
77	45, 76	rexlimddv 2599	. 2 ⊢ (𝜑 → ∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵))
78		eleq1w 2238	. . 3 ⊢ (𝑗 = 𝑠 → (𝑗 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ (2nd ‘𝐵)))
79	78	cbvrexv 2704	. 2 ⊢ (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))
80	77, 79	sylib 122	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459  Vcvv 2737   ⊆ wss 3129  ⟨cop 3595  ∪ cuni 3809  ∩ cint 3844   class class class wbr 4003   Or wor 4295   “ cima 4629  Fun wfun 5210  –onto→wfo 5214  ‘cfv 5216  (class class class)co 5874  1^st c1st 6138  2^nd c2nd 6139  Qcnq 7278  1_Qc1q 7279   +_Q cplq 7280   <_Q cltq 7283  Pcnp 7289  <_P cltp 7293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iltp 7468

This theorem is referenced by:  suplocexprlemex  7720