Theorem ipodrsima 16988

Description: The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Hypotheses

Ref	Expression
ipodrsima.f	⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴)
ipodrsima.m	⊢ ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣))
ipodrsima.d	⊢ (𝜑 → (toInc‘𝐵) ∈ Dirset)
ipodrsima.s	⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝐴)
ipodrsima.a	⊢ (𝜑 → (𝐹 “ 𝐵) ∈ 𝑉)

Assertion

Ref	Expression
ipodrsima	⊢ (𝜑 → (toInc‘(𝐹 “ 𝐵)) ∈ Dirset)

Distinct variable groups:   𝜑,𝑢,𝑣   𝑢,𝐴,𝑣   𝑢,𝐹,𝑣

Allowed substitution hints:   𝐵(𝑣,𝑢)   𝑉(𝑣,𝑢)

Proof of Theorem ipodrsima

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipodrsima.a	. . 3 ⊢ (𝜑 → (𝐹 “ 𝐵) ∈ 𝑉)
2		elex 3185	. . 3 ⊢ ((𝐹 “ 𝐵) ∈ 𝑉 → (𝐹 “ 𝐵) ∈ V)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹 “ 𝐵) ∈ V)
4		ipodrsima.d	. . . . 5 ⊢ (𝜑 → (toInc‘𝐵) ∈ Dirset)
5		isipodrs 16984	. . . . 5 ⊢ ((toInc‘𝐵) ∈ Dirset ↔ (𝐵 ∈ V ∧ 𝐵 ≠ ∅ ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐))
6	4, 5	sylib 207	. . . 4 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐵 ≠ ∅ ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐))
7	6	simp2d 1067	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
8		ipodrsima.f	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴)
9		ipodrsima.s	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝐴)
10		fnimaeq0 5926	. . . . 5 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ 𝐵 ⊆ 𝒫 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))
11	8, 9, 10	syl2anc 691	. . . 4 ⊢ (𝜑 → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))
12	11	necon3bid 2826	. . 3 ⊢ (𝜑 → ((𝐹 “ 𝐵) ≠ ∅ ↔ 𝐵 ≠ ∅))
13	7, 12	mpbird 246	. 2 ⊢ (𝜑 → (𝐹 “ 𝐵) ≠ ∅)
14	6	simp3d 1068	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐)
15		simplll 794	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑎 ⊆ 𝑐) → 𝜑)
16		simpr 476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑎 ⊆ 𝑐) → 𝑎 ⊆ 𝑐)
17	9	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐵 ⊆ 𝒫 𝐴)
18		simprr 792	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
19	17, 18	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝒫 𝐴)
20	19	elpwid 4118	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ⊆ 𝐴)
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑎 ⊆ 𝑐) → 𝑐 ⊆ 𝐴)
22		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
23		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ V
24		sseq12 3591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → (𝑢 ⊆ 𝑣 ↔ 𝑎 ⊆ 𝑐))
25		sseq1 3589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑐 → (𝑣 ⊆ 𝐴 ↔ 𝑐 ⊆ 𝐴))
26	25	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → (𝑣 ⊆ 𝐴 ↔ 𝑐 ⊆ 𝐴))
27	24, 26	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → ((𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑎 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)))
28	27	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) ↔ (𝜑 ∧ (𝑎 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴))))
29		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑎 → (𝐹‘𝑢) = (𝐹‘𝑎))
30		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑐 → (𝐹‘𝑣) = (𝐹‘𝑐))
31		sseq12 3591	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑢) = (𝐹‘𝑎) ∧ (𝐹‘𝑣) = (𝐹‘𝑐)) → ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ↔ (𝐹‘𝑎) ⊆ (𝐹‘𝑐)))
32	29, 30, 31	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ↔ (𝐹‘𝑎) ⊆ (𝐹‘𝑐)))
33	28, 32	imbi12d 333	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑎 ∧ 𝑣 = 𝑐) → (((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣)) ↔ ((𝜑 ∧ (𝑎 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑐))))
34		ipodrsima.m	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣))
35	22, 23, 33, 34	vtocl2 3234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑐))
36	15, 16, 21, 35	syl12anc 1316	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑎 ⊆ 𝑐) → (𝐹‘𝑎) ⊆ (𝐹‘𝑐))
37	36	ex 449	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎 ⊆ 𝑐 → (𝐹‘𝑎) ⊆ (𝐹‘𝑐)))
38		simplll 794	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑏 ⊆ 𝑐) → 𝜑)
39		simpr 476	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑏 ⊆ 𝑐) → 𝑏 ⊆ 𝑐)
40	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑏 ⊆ 𝑐) → 𝑐 ⊆ 𝐴)
41		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
42		sseq12 3591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → (𝑢 ⊆ 𝑣 ↔ 𝑏 ⊆ 𝑐))
43	25	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → (𝑣 ⊆ 𝐴 ↔ 𝑐 ⊆ 𝐴))
44	42, 43	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → ((𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑏 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)))
45	44	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) ↔ (𝜑 ∧ (𝑏 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴))))
46		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑏 → (𝐹‘𝑢) = (𝐹‘𝑏))
47		sseq12 3591	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑢) = (𝐹‘𝑏) ∧ (𝐹‘𝑣) = (𝐹‘𝑐)) → ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ↔ (𝐹‘𝑏) ⊆ (𝐹‘𝑐)))
48	46, 30, 47	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ↔ (𝐹‘𝑏) ⊆ (𝐹‘𝑐)))
49	45, 48	imbi12d 333	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑏 ∧ 𝑣 = 𝑐) → (((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣)) ↔ ((𝜑 ∧ (𝑏 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑐))))
50	41, 23, 49, 34	vtocl2 3234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑐 ∧ 𝑐 ⊆ 𝐴)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑐))
51	38, 39, 40, 50	syl12anc 1316	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ 𝑏 ⊆ 𝑐) → (𝐹‘𝑏) ⊆ (𝐹‘𝑐))
52	51	ex 449	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏 ⊆ 𝑐 → (𝐹‘𝑏) ⊆ (𝐹‘𝑐)))
53	37, 52	anim12d 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐) → ((𝐹‘𝑎) ⊆ (𝐹‘𝑐) ∧ (𝐹‘𝑏) ⊆ (𝐹‘𝑐))))
54		unss 3749	. . . . . . . . 9 ⊢ ((𝑎 ⊆ 𝑐 ∧ 𝑏 ⊆ 𝑐) ↔ (𝑎 ∪ 𝑏) ⊆ 𝑐)
55		unss 3749	. . . . . . . . 9 ⊢ (((𝐹‘𝑎) ⊆ (𝐹‘𝑐) ∧ (𝐹‘𝑏) ⊆ (𝐹‘𝑐)) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
56	53, 54, 55	3imtr3g 283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 ∪ 𝑏) ⊆ 𝑐 → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
57	56	anassrs 678	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → ((𝑎 ∪ 𝑏) ⊆ 𝑐 → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
58	57	reximdva 3000	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐 → ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
59	58	ralimdva 2945	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐 → ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
60	59	ralimdva 2945	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 (𝑎 ∪ 𝑏) ⊆ 𝑐 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
61	14, 60	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
62		uneq1 3722	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 ∪ 𝑦) = ((𝐹‘𝑎) ∪ 𝑦))
63	62	sseq1d 3595	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧))
64	63	rexbidv 3034	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧))
65	64	ralbidv 2969	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧))
66	65	ralima 6402	. . . . 5 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ 𝐵 ⊆ 𝒫 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑎 ∈ 𝐵 ∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧))
67	8, 9, 66	syl2anc 691	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (𝐹 “ 𝐵)∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑎 ∈ 𝐵 ∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧))
68		uneq2 3723	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑦) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
69	68	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧))
70	69	rexbidv 3034	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧))
71	70	ralima 6402	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ 𝐵 ⊆ 𝒫 𝐴) → (∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑏 ∈ 𝐵 ∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧))
72	8, 9, 71	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑏 ∈ 𝐵 ∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧))
73		sseq2 3590	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
74	73	rexima 6401	. . . . . . . 8 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ 𝐵 ⊆ 𝒫 𝐴) → (∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧 ↔ ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
75	8, 9, 74	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧 ↔ ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	ralbidv 2969	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝐵 ∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑧 ↔ ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	72, 76	bitrd 267	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
78	77	ralbidv 2969	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝐵 ∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)((𝐹‘𝑎) ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
79	67, 78	bitrd 267	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (𝐹 “ 𝐵)∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧 ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐵 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
80	61, 79	mpbird 246	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝐵)∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧)
81		isipodrs 16984	. 2 ⊢ ((toInc‘(𝐹 “ 𝐵)) ∈ Dirset ↔ ((𝐹 “ 𝐵) ∈ V ∧ (𝐹 “ 𝐵) ≠ ∅ ∧ ∀𝑥 ∈ (𝐹 “ 𝐵)∀𝑦 ∈ (𝐹 “ 𝐵)∃𝑧 ∈ (𝐹 “ 𝐵)(𝑥 ∪ 𝑦) ⊆ 𝑧))
82	3, 13, 80, 81	syl3anbrc 1239	1 ⊢ (𝜑 → (toInc‘(𝐹 “ 𝐵)) ∈ Dirset)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   “ cima 5041   Fn wfn 5799  ‘cfv 5804  Dirsetcdrs 16750  toInccipo 16974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-tset 15787  df-ple 15788  df-ocomp 15790  df-preset 16751  df-drs 16752  df-poset 16769  df-ipo 16975

This theorem is referenced by:  isacs4lem  16991  isnacs3  36291