Theorem bnj1118 33596

Description: Technical lemma for bnj69 33622. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1118.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1118.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1118.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1118.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1118.18	⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
bnj1118.19	⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
bnj1118.26	⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Assertion

Ref	Expression
bnj1118	⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Distinct variable groups:   𝐴,𝑖,𝑗,𝑦   𝑦,𝐵   𝐷,𝑗   𝑅,𝑖,𝑗,𝑦   𝑓,𝑖,𝑗,𝑦   𝑖,𝑛,𝑗

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜏(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑓,𝑛)   𝐵(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑖,𝑛)   𝑅(𝑓,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜑₀(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1118

Step	Hyp	Ref	Expression
1		bnj1118.3	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj1118.7	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
3		bnj1118.18	. . . 4 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
4		bnj1118.19	. . . 4 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
5		bnj1118.26	. . . 4 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
6	1, 2, 3, 4, 5	bnj1110 33594	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))
7		ancl 545	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))))
8	6, 7	bnj101 33335	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)))
9		simpr2 1195	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
10	1	bnj1254 33421	. . . . . . 7 ⊢ (𝜒 → 𝜓)
11	10	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝜓)
12	11	ad2antrl 726	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜓)
13	12	adantr 481	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝜓)
14	1	bnj1232 33415	. . . . . . . . 9 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
15	14	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
16	15	ad2antrl 726	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑛 ∈ 𝐷)
17	16	adantr 481	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑛 ∈ 𝐷)
18		simpr1 1194	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ 𝑛)
19	2	bnj923 33380	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
20	19	anim1i 615	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑛 ∈ ω ∧ 𝑗 ∈ 𝑛))
21	20	ancomd 462	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
22	17, 18, 21	syl2anc 584	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
23		elnn 7813	. . . . 5 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω)
24	22, 23	syl 17	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
25	4	bnj1232 33415	. . . . . 6 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
26	25	adantl 482	. . . . 5 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → 𝑖 ∈ 𝑛)
27	26	ad2antlr 725	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 ∈ 𝑛)
28	9, 13, 24, 27	bnj951 33387	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛))
29		bnj1118.5	. . . . . . 7 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
30	29	simp2bi 1146	. . . . . 6 ⊢ (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31	30	3ad2ant2 1134	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → TrFo(𝐵, 𝐴, 𝑅))
32	31	ad2antrl 726	. . . 4 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33		simp3 1138	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵) → (𝑓‘𝑗) ⊆ 𝐵)
34	32, 33	anim12i 613	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵))
35		bnj256 33318	. . . . 5 ⊢ ((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ↔ ((𝑖 = suc 𝑗 ∧ 𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛)))
36		bnj1118.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
37	36	bnj1112 33595	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
38	37	biimpi 215	. . . . . . . 8 ⊢ (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
39	38	19.21bi 2182	. . . . . . 7 ⊢ (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
40		eleq1 2825	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
41	40	anbi2d 629	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
42		fveqeq2 6851	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
43	41, 42	imbi12d 344	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
44	39, 43	syl5ibr 245	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
45	44	imp31 418	. . . . 5 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
46	35, 45	sylbi 216	. . . 4 ⊢ ((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
47		df-bnj19 33309	. . . . . . 7 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
48		ssralv 4010	. . . . . . 7 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
49	47, 48	biimtrid 241	. . . . . 6 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
50	49	impcom 408	. . . . 5 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
51		iunss 5005	. . . . 5 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
52	50, 51	sylibr 233	. . . 4 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53		sseq1 3969	. . . . 5 ⊢ ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓‘𝑖) ⊆ 𝐵 ↔ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
54	53	biimpar 478	. . . 4 ⊢ (((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ∧ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵)
55	46, 52, 54	syl2an 596	. . 3 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
56	28, 34, 55	syl2anc 584	. 2 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
57	8, 56	bnj1023 33392	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  ∅c0 4282  {csn 4586  ∪ ciun 4954   class class class wbr 5105   E cep 5536  dom cdm 5633  suc csuc 6319   Fn wfn 6491  ‘cfv 6496  ωcom 7802   ∧ w-bnj17 33298   predc-bnj14 33300   TrFow-bnj19 33308

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fn 6499  df-fv 6504  df-om 7803  df-bnj17 33299  df-bnj19 33309

This theorem is referenced by:  bnj1030  33599