Theorem bnj1118 35614

Description: Technical lemma for bnj69 35640. 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 35612	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))
7		ancl 554	. . 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 35354	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)))
9		simpr2 1214	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
10	1	bnj1254 35439	. . . . . . 7 ⊢ (𝜒 → 𝜓)
11	10	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝜓)
12	11	ad2antrl 741	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜓)
13	12	adantr 486	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝜓)
14	1	bnj1232 35433	. . . . . . . . 9 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
15	14	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
16	15	ad2antrl 741	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑛 ∈ 𝐷)
17	16	adantr 486	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑛 ∈ 𝐷)
18		simpr1 1213	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ 𝑛)
19	2	bnj923 35399	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
20	19	anim1i 627	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑛 ∈ ω ∧ 𝑗 ∈ 𝑛))
21	20	ancomd 467	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
22	17, 18, 21	syl2anc 596	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
23		elnn 7888	. . . . 5 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω)
24	22, 23	syl 18	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
25	4	bnj1232 35433	. . . . . 6 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
26	25	adantl 487	. . . . 5 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → 𝑖 ∈ 𝑛)
27	26	ad2antlr 740	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 ∈ 𝑛)
28	9, 13, 24, 27	bnj951 35406	. . 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 1164	. . . . . 6 ⊢ (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31	30	3ad2ant2 1152	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → TrFo(𝐵, 𝐴, 𝑅))
32	31	ad2antrl 741	. . . 4 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33		simp3 1156	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵) → (𝑓‘𝑗) ⊆ 𝐵)
34	32, 33	anim12i 625	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵))
35		bnj256 35337	. . . . 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 35613	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
38	37	biimpi 219	. . . . . . . 8 ⊢ (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
39	38	19.21bi 2225	. . . . . . 7 ⊢ (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
40		eleq1 2848	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
41	40	anbi2d 642	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
42		fveqeq2 6894	. . . . . . . 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 347	. . . . . . 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	imbitrrid 249	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
45	44	imp31 423	. . . . 5 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
46	35, 45	sylbi 220	. . . 4 ⊢ ((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
47		df-bnj19 35328	. . . . . . 7 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
48		ssralv 3977	. . . . . . 7 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
49	47, 48	biimtrid 245	. . . . . 6 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
50	49	impcom 413	. . . . 5 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
51		iunss 5002	. . . . 5 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
52	50, 51	sylibr 237	. . . 4 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53		sseq1 3933	. . . . 5 ⊢ ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓‘𝑖) ⊆ 𝐵 ↔ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
54	53	biimpar 483	. . . 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 608	. . 3 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
56	28, 34, 55	syl2anc 596	. 2 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
57	8, 56	bnj1023 35411	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  {csn 4583  ∪ ciun 4950   class class class wbr 5102   E cep 5549  dom cdm 5650  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   ∧ w-bnj17 35317   predc-bnj14 35319   TrFow-bnj19 35327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fn 6541  df-fv 6546  df-om 7878  df-bnj17 35318  df-bnj19 35328

This theorem is used by:  bnj1030  35617