Theorem bnj1128 32372

Description: Technical lemma for bnj69 32392. 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
bnj1128.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1128.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1128.3	⊢ 𝐷 = (ω ∖ {∅})
bnj1128.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1128.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1128.6	⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))
bnj1128.7	⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
bnj1128.8	⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)
bnj1128.9	⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
bnj1128.10	⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)
bnj1128.11	⊢ (𝜃′ ↔ [𝑗 / 𝑖]𝜃)

Assertion

Ref	Expression
bnj1128	⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑛,𝑦   𝐷,𝑖,𝑗,𝑦   𝑅,𝑓,𝑖,𝑗,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑌,𝑖,𝑛,𝑦   𝜒,𝑗   𝜑,𝑖,𝑦   𝜃,𝑗

Allowed substitution hints:   𝜑(𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑛)   𝜏(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑛)   𝑋(𝑗)   𝑌(𝑗)   𝜑′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜃′(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1128

Step	Hyp	Ref	Expression
1		bnj1128.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1128.2	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1128.3	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj1128.4	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5		bnj1128.5	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	1, 2, 3, 4, 5	bnj981 32332	. . 3 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)))
7		simp1 1133	. . . . . 6 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝜒)
8		simp2 1134	. . . . . 6 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑖 ∈ 𝑛)
9		bnj1128.7	. . . . . . . . 9 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
10		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑛
11		nfra1 3183	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)
12	9, 11	nfxfr 1854	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜏
13		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜒
14	10, 12, 13	nf3an 1902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)
15		nfv 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑓‘𝑖) ⊆ 𝐴
16	14, 15	nfim 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
17	16	nf5ri 2193	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) → ∀𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴))
18	3	bnj1098 32165	. . . . . . . . . . . . . . . . 17 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
19		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ≠ ∅)
20		simpr1 1191	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ∈ 𝑛)
21	5	bnj1232 32185	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
22	21	3ad2ant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
23	22	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑛 ∈ 𝐷)
24	19, 20, 23	3jca 1125	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
25	18, 24	bnj1101 32166	. . . . . . . . . . . . . . . 16 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
26		ancl 548	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
27	25, 26	bnj101 32103	. . . . . . . . . . . . . . 15 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))
28		df-3an 1086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))
29	28	imbi2i 339	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
30	29	exbii 1849	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))))
31	27, 30	mpbir 234	. . . . . . . . . . . . . 14 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))
32		bnj213 32264	. . . . . . . . . . . . . . . 16 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
33	32	bnj226 32114	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
34		simp21 1203	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 ∈ 𝑛)
35		simp3r 1199	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗)
36		biid 264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷)
37		biid 264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛)
38		bnj1128.8	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)
39		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑗 ∈ V
40		sbcg 3793	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑 ↔ 𝜑))
41	39, 40	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑗 / 𝑖]𝜑 ↔ 𝜑)
42	38, 41	bitr2i 279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 ↔ 𝜑′)
43		bnj1128.9	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
44	2, 43	bnj1039 32353	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
45	2, 44	bitr4i 281	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜓 ↔ 𝜓′)
46	36, 37, 42, 45	bnj887 32146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
47		bnj1128.10	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)
48	38, 43, 5, 47	bnj1040 32354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
49	46, 5, 48	3bitr4i 306	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 ↔ 𝜒′)
50	48	bnj1254 32191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒′ → 𝜓′)
51	49, 50	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝜓′)
52	51	3ad2ant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜓′)
53	52	3ad2ant2 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝜓′)
54		simp3l 1198	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ 𝑛)
55	22	3ad2ant2 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑛 ∈ 𝐷)
56	3	bnj923 32149	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
57		elnn 7570	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω)
58	56, 57	sylan2 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → 𝑗 ∈ ω)
59	54, 55, 58	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ ω)
60	44	bnj589 32291	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
61		rsp 3170	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
62	60, 61	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜓′ → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
63		eleq1 2877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
64		fveqeq2 6654	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
65	63, 64	imbi12d 348	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = suc 𝑗 → ((𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
66	65	imbi2d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (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)))))
67	62, 66	syl5ibr 249	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = suc 𝑗 → (𝜓′ → (𝑗 ∈ ω → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))))
68	35, 53, 59, 67	syl3c 66	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
69	34, 68	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
70	33, 69	bnj1262 32192	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) ⊆ 𝐴)
71	31, 70	bnj1023 32162	. . . . . . . . . . . . 13 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴)
72	5	bnj1247 32190	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → 𝜑)
73	72	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜑)
74		bnj213 32264	. . . . . . . . . . . . . . 15 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅))
76	1	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
77	75, 76	sylan9eq 2853	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅))
78	74, 77	bnj1262 32192	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) ⊆ 𝐴)
79	73, 78	sylan2 595	. . . . . . . . . . . . 13 ⊢ ((𝑖 = ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴)
80	71, 79	bnj1109 32168	. . . . . . . . . . . 12 ⊢ ∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
81	17, 80	bnj1131 32169	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)
82	81	3expia 1118	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))
83		bnj1128.6	. . . . . . . . . 10 ⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))
84	82, 83	sylibr 237	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
85	3, 5, 9, 84	bnj1133 32371	. . . . . . . 8 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)
86	83	ralbii 3133	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑛 𝜃 ↔ ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))
87	85, 86	sylib 221	. . . . . . 7 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))
88		rsp 3170	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴) → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)))
89	87, 88	syl 17	. . . . . 6 ⊢ (𝜒 → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)))
90	7, 8, 7, 89	syl3c 66	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → (𝑓‘𝑖) ⊆ 𝐴)
91		simp3 1135	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ (𝑓‘𝑖))
92	90, 91	sseldd 3916	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ 𝐴)
93	92	2eximi 1837	. . 3 ⊢ (∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → ∃𝑛∃𝑖 𝑌 ∈ 𝐴)
94	6, 93	bnj593 32126	. 2 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴)
95		19.9v 1988	. . 3 ⊢ (∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑛∃𝑖 𝑌 ∈ 𝐴)
96		19.9v 1988	. . 3 ⊢ (∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑖 𝑌 ∈ 𝐴)
97		19.9v 1988	. . 3 ⊢ (∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴)
98	95, 96, 97	3bitri 300	. 2 ⊢ (∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴)
99	94, 98	sylib 221	1 ⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441  [wsbc 3720   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∪ ciun 4881   class class class wbr 5030   E cep 5429  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  ωcom 7560   ∧ w-bnj17 32066   predc-bnj14 32068   trClc-bnj18 32074

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fn 6327  df-fv 6332  df-om 7561  df-bnj17 32067  df-bnj14 32069  df-bnj18 32075

This theorem is referenced by:  bnj1127  32373