Theorem bnj1398 33646

Description: Technical lemma for bnj60 33674. 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
bnj1398.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1398.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1398.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1398.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1398.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1398.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1398.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1398.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1398.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1398.10	⊢ 𝑃 = ∪ 𝐻
bnj1398.11	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
bnj1398.12	⊢ (𝜂 ↔ (𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

Assertion

Ref	Expression
bnj1398	⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐶   𝑦,𝐷   𝑧,𝐻   𝑧,𝑃   𝑅,𝑓,𝑥,𝑦,𝑧   𝜒,𝑧   𝑓,𝑑,𝑥   𝜓,𝑦   𝜏,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑑)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1398

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1398.11	. . . . 5 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
2		df-iun 4956	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = {𝑧 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))}
3	2	bnj1436 33451	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
4	1, 3	simplbiim 505	. . . . . . . 8 ⊢ (𝜃 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
5		bnj1398.12	. . . . . . . 8 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
6		bnj1398.7	. . . . . . . . . . . 12 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
7		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝜓
8		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
9		nfra1 3267	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
10	7, 8, 9	nf3an 1904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
11	6, 10	nfxfr 1855	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜒
12		nfiu1 4988	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
13	12	nfcri 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
14	11, 13	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
15	1, 14	nfxfr 1855	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜃
16	15	nf5ri 2188	. . . . . . . 8 ⊢ (𝜃 → ∀𝑦𝜃)
17	4, 5, 16	bnj1521 33463	. . . . . . 7 ⊢ (𝜃 → ∃𝑦𝜂)
18		bnj1398.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
19		nfv 1917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑓 𝑅 FrSe 𝐴
20		bnj1398.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
21		nfe1 2147	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑓∃𝑓𝜏
22	21	nfn 1860	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓 ¬ ∃𝑓𝜏
23		nfcv 2907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓𝐴
24	22, 23	nfrabw 3440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓{𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
25	20, 24	nfcxfr 2905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑓𝐷
26		nfcv 2907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑓∅
27	25, 26	nfne 3045	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑓 𝐷 ≠ ∅
28	19, 27	nfan 1902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)
29	18, 28	nfxfr 1855	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓𝜓
30	25	nfcri 2894	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓 𝑥 ∈ 𝐷
31		nfv 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑓 ¬ 𝑦𝑅𝑥
32	25, 31	nfralw 3294	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
33	29, 30, 32	nf3an 1904	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑓(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
34	6, 33	nfxfr 1855	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑓𝜒
35		nfv 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑓 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
36	34, 35	nfan 1902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑓(𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
37	1, 36	nfxfr 1855	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓𝜃
38		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)
39		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
40	37, 38, 39	nf3an 1904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓(𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
41	5, 40	nfxfr 1855	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝜂
42	41	nf5ri 2188	. . . . . . . . . . 11 ⊢ (𝜂 → ∀𝑓𝜂)
43	1	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝜃 → 𝜒)
44	5, 43	bnj835 33371	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝜒)
45	5	simp2bi 1146	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
46		bnj1398.1	. . . . . . . . . . . . . 14 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
47		bnj1398.2	. . . . . . . . . . . . . 14 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
48		bnj1398.3	. . . . . . . . . . . . . 14 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
49		bnj1398.4	. . . . . . . . . . . . . 14 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
50		bnj1398.8	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
51	46, 47, 48, 49, 20, 18, 6, 50	bnj1388 33645	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
52		rsp 3230	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′ → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
54	44, 45, 53	sylc 65	. . . . . . . . . . 11 ⊢ (𝜂 → ∃𝑓𝜏′)
55	42, 54	bnj596 33358	. . . . . . . . . 10 ⊢ (𝜂 → ∃𝑓(𝜂 ∧ 𝜏′))
56	46, 47, 48, 49, 50	bnj1373 33642	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
57	56	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝜏′ → 𝑓 ∈ 𝐶)
58	57	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → 𝑓 ∈ 𝐶)
59	56	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝜏′ → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
60		rspe 3232	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
61	45, 59, 60	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
62		bnj1398.9	. . . . . . . . . . . . . 14 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
63	62	eqabi 2881	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐻 ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
64	56	rexbii 3097	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
65		r19.42v 3187	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
66	63, 64, 65	3bitri 296	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐻 ↔ (𝑓 ∈ 𝐶 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
67	58, 61, 66	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((𝜂 ∧ 𝜏′) → 𝑓 ∈ 𝐻)
68	5	simp3bi 1147	. . . . . . . . . . . . 13 ⊢ (𝜂 → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
69	68	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
70	59	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
71	69, 70	eleqtrrd 2841	. . . . . . . . . . 11 ⊢ ((𝜂 ∧ 𝜏′) → 𝑧 ∈ dom 𝑓)
72	67, 71	jca 512	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜏′) → (𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
73	55, 72	bnj593 33357	. . . . . . . . 9 ⊢ (𝜂 → ∃𝑓(𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
74		df-rex 3074	. . . . . . . . 9 ⊢ (∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓 ↔ ∃𝑓(𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
75	73, 74	sylibr 233	. . . . . . . 8 ⊢ (𝜂 → ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
76		bnj1398.10	. . . . . . . . . . . 12 ⊢ 𝑃 = ∪ 𝐻
77	76	dmeqi 5860	. . . . . . . . . . 11 ⊢ dom 𝑃 = dom ∪ 𝐻
78	62	bnj1317 33433	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
79	78	bnj1400 33447	. . . . . . . . . . 11 ⊢ dom ∪ 𝐻 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
80	77, 79	eqtri 2764	. . . . . . . . . 10 ⊢ dom 𝑃 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
81	80	eleq2i 2829	. . . . . . . . 9 ⊢ (𝑧 ∈ dom 𝑃 ↔ 𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓)
82		eliun 4958	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓 ↔ ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
83	81, 82	bitri 274	. . . . . . . 8 ⊢ (𝑧 ∈ dom 𝑃 ↔ ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
84	75, 83	sylibr 233	. . . . . . 7 ⊢ (𝜂 → 𝑧 ∈ dom 𝑃)
85	17, 84	bnj593 33357	. . . . . 6 ⊢ (𝜃 → ∃𝑦 𝑧 ∈ dom 𝑃)
86		nfre1 3268	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
87	86	nfab 2913	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
88	62, 87	nfcxfr 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐻
89	88	nfuni 4872	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝐻
90	76, 89	nfcxfr 2905	. . . . . . . 8 ⊢ Ⅎ𝑦𝑃
91	90	nfdm 5906	. . . . . . 7 ⊢ Ⅎ𝑦dom 𝑃
92	91	nfcrii 2899	. . . . . 6 ⊢ (𝑧 ∈ dom 𝑃 → ∀𝑦 𝑧 ∈ dom 𝑃)
93	85, 92	bnj1397 33446	. . . . 5 ⊢ (𝜃 → 𝑧 ∈ dom 𝑃)
94	1, 93	sylbir 234	. . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ dom 𝑃)
95	94	ex 413	. . 3 ⊢ (𝜒 → (𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ dom 𝑃))
96	95	ssrdv 3950	. 2 ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ⊆ dom 𝑃)
97		simpr 485	. . . . . . 7 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
98	66	simprbi 497	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
99	98	adantr 481	. . . . . . 7 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
100		r19.42v 3187	. . . . . . . 8 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑧 ∈ dom 𝑓 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
101		eleq2 2826	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
102	101	biimpac 479	. . . . . . . . 9 ⊢ ((𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
103	102	reximi 3087	. . . . . . . 8 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
104	100, 103	sylbir 234	. . . . . . 7 ⊢ ((𝑧 ∈ dom 𝑓 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
105	97, 99, 104	syl2anc 584	. . . . . 6 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
106	105	rexlimiva 3144	. . . . 5 ⊢ (∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
107		eliun 4958	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
108	106, 83, 107	3imtr4i 291	. . . 4 ⊢ (𝑧 ∈ dom 𝑃 → 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
109	108	ssriv 3948	. . 3 ⊢ dom 𝑃 ⊆ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
110	109	a1i 11	. 2 ⊢ (𝜒 → dom 𝑃 ⊆ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
111	96, 110	eqssd 3961	1 ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  [wsbc 3739   ∪ cun 3908   ⊆ wss 3910  ∅c0 4282  {csn 4586  ⟨cop 4592  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105  dom cdm 5633   ↾ cres 5635   Fn wfn 6491  ‘cfv 6496   predc-bnj14 33300   FrSe w-bnj15 33304   trClc-bnj18 33306

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

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-dm 5643  df-bnj14 33301  df-bnj18 33307

This theorem is referenced by:  bnj1415  33650