Theorem bnj1398 35367

Description: Technical lemma for bnj60 35395. 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 4960	. . . . . . . . . 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 35172	. . . . . . . . 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 513	. . . . . . . 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 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝜓
8		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
9		nfra1 3295	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
10	7, 8, 9	nf3an 1928	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
11	6, 10	nfxfr 1880	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜒
12		nfiu1 4994	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
13	12	nfcri 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
14	11, 13	nfan 1926	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
15	1, 14	nfxfr 1880	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜃
16	15	nf5ri 2237	. . . . . . . 8 ⊢ (𝜃 → ∀𝑦𝜃)
17	4, 5, 16	bnj1521 35184	. . . . . . 7 ⊢ (𝜃 → ∃𝑦𝜂)
18		bnj1398.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
19		nfv 1941	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑓 𝑅 FrSe 𝐴
20		bnj1398.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
21		nfe1 2191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑓∃𝑓𝜏
22	21	nfn 1884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓 ¬ ∃𝑓𝜏
23		nfcv 2931	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓𝐴
24	22, 23	nfrabw 3458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓{𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
25	20, 24	nfcxfr 2929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑓𝐷
26		nfcv 2931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑓∅
27	25, 26	nfne 3067	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑓 𝐷 ≠ ∅
28	19, 27	nfan 1926	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)
29	18, 28	nfxfr 1880	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓𝜓
30	25	nfcri 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓 𝑥 ∈ 𝐷
31		nfv 1941	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑓 ¬ 𝑦𝑅𝑥
32	25, 31	nfralw 3318	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑓∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
33	29, 30, 32	nf3an 1928	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑓(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
34	6, 33	nfxfr 1880	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑓𝜒
35		nfv 1941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑓 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
36	34, 35	nfan 1926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑓(𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
37	1, 36	nfxfr 1880	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓𝜃
38		nfv 1941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)
39		nfv 1941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
40	37, 38, 39	nf3an 1928	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓(𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
41	5, 40	nfxfr 1880	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝜂
42	41	nf5ri 2237	. . . . . . . . . . 11 ⊢ (𝜂 → ∀𝑓𝜂)
43	1	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝜃 → 𝜒)
44	5, 43	bnj835 35093	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝜒)
45	5	simp2bi 1162	. . . . . . . . . . . 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 35366	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
52		rsp 3259	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′ → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
53	51, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
54	44, 45, 53	sylc 66	. . . . . . . . . . 11 ⊢ (𝜂 → ∃𝑓𝜏′)
55	42, 54	bnj596 35080	. . . . . . . . . 10 ⊢ (𝜂 → ∃𝑓(𝜂 ∧ 𝜏′))
56	46, 47, 48, 49, 50	bnj1373 35363	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
57	56	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝜏′ → 𝑓 ∈ 𝐶)
58	57	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → 𝑓 ∈ 𝐶)
59	56	simprbi 502	. . . . . . . . . . . . 13 ⊢ (𝜏′ → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
60		rspe 3261	. . . . . . . . . . . . 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 607	. . . . . . . . . . . 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	eqabri 2911	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐻 ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
64	56	rexbii 3118	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
65		r19.42v 3203	. . . . . . . . . . . . 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 300	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐻 ↔ (𝑓 ∈ 𝐶 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
67	58, 61, 66	sylanbrc 594	. . . . . . . . . . 11 ⊢ ((𝜂 ∧ 𝜏′) → 𝑓 ∈ 𝐻)
68	5	simp3bi 1163	. . . . . . . . . . . . 13 ⊢ (𝜂 → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
69	68	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
70	59	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ 𝜏′) → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
71	69, 70	eleqtrrd 2872	. . . . . . . . . . 11 ⊢ ((𝜂 ∧ 𝜏′) → 𝑧 ∈ dom 𝑓)
72	67, 71	jca 520	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜏′) → (𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
73	55, 72	bnj593 35079	. . . . . . . . 9 ⊢ (𝜂 → ∃𝑓(𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
74		df-rex 3096	. . . . . . . . 9 ⊢ (∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓 ↔ ∃𝑓(𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
75	73, 74	sylibr 237	. . . . . . . 8 ⊢ (𝜂 → ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
76		bnj1398.10	. . . . . . . . . . . 12 ⊢ 𝑃 = ∪ 𝐻
77	76	dmeqi 5895	. . . . . . . . . . 11 ⊢ dom 𝑃 = dom ∪ 𝐻
78	62	bnj1317 35154	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
79	78	bnj1400 35168	. . . . . . . . . . 11 ⊢ dom ∪ 𝐻 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
80	77, 79	eqtri 2792	. . . . . . . . . 10 ⊢ dom 𝑃 = ∪ 𝑓 ∈ 𝐻 dom 𝑓
81	80	eleq2i 2861	. . . . . . . . 9 ⊢ (𝑧 ∈ dom 𝑃 ↔ 𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓)
82		eliun 4962	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐻 dom 𝑓 ↔ ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
83	81, 82	bitri 278	. . . . . . . 8 ⊢ (𝑧 ∈ dom 𝑃 ↔ ∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓)
84	75, 83	sylibr 237	. . . . . . 7 ⊢ (𝜂 → 𝑧 ∈ dom 𝑃)
85	17, 84	bnj593 35079	. . . . . 6 ⊢ (𝜃 → ∃𝑦 𝑧 ∈ dom 𝑃)
86		nfre1 3296	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
87	86	nfab 2937	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
88	62, 87	nfcxfr 2929	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐻
89	88	nfuni 4881	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝐻
90	76, 89	nfcxfr 2929	. . . . . . . 8 ⊢ Ⅎ𝑦𝑃
91	90	nfdm 5942	. . . . . . 7 ⊢ Ⅎ𝑦dom 𝑃
92	91	nfcrii 2926	. . . . . 6 ⊢ (𝑧 ∈ dom 𝑃 → ∀𝑦 𝑧 ∈ dom 𝑃)
93	85, 92	bnj1397 35167	. . . . 5 ⊢ (𝜃 → 𝑧 ∈ dom 𝑃)
94	1, 93	sylbir 238	. . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ dom 𝑃)
95	94	ex 417	. . 3 ⊢ (𝜒 → (𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ dom 𝑃))
96	95	ssrdv 3949	. 2 ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ⊆ dom 𝑃)
97		simpr 489	. . . . . . 7 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
98	66	simprbi 502	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
99	98	adantr 485	. . . . . . 7 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
100		r19.42v 3203	. . . . . . . 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 2858	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
102	101	biimpac 483	. . . . . . . . 9 ⊢ ((𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
103	102	reximi 3109	. . . . . . . 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 238	. . . . . . 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 595	. . . . . 6 ⊢ ((𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
106	105	rexlimiva 3164	. . . . 5 ⊢ (∃𝑓 ∈ 𝐻 𝑧 ∈ dom 𝑓 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
107		eliun 4962	. . . . 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 295	. . . 4 ⊢ (𝑧 ∈ dom 𝑃 → 𝑧 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
109	108	ssriv 3947	. . 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 3960	1 ⊢ (𝜒 → ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3422  [wsbc 3751   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4592  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111  dom cdm 5662   ↾ cres 5664   Fn wfn 6532  ‘cfv 6537   predc-bnj14 35022   FrSe w-bnj15 35026   trClc-bnj18 35028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-dm 5672  df-bnj14 35023  df-bnj18 35029

This theorem is referenced by:  bnj1415  35371