Theorem bnj1523 34638

Description: Technical lemma for bnj1522 34639. 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
bnj1523.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1523.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1523.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1523.4	⊢ 𝐹 = ∪ 𝐶
bnj1523.5	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
bnj1523.6	⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))
bnj1523.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))
bnj1523.8	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}
bnj1523.9	⊢ (𝜃 ↔ (𝜒 ∧ 𝑦 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦))

Assertion

Ref	Expression
bnj1523	⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝑦,𝐴,𝑧,𝑥   𝐵,𝑓   𝑦,𝐷,𝑧   𝑦,𝐹,𝑧   𝐺,𝑑,𝑓,𝑥   𝑦,𝐺   𝑥,𝐻,𝑦,𝑧   𝑅,𝑑,𝑓,𝑥   𝑦,𝑅,𝑧   𝑌,𝑑   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑧)   𝐻(𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓)

Proof of Theorem bnj1523

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1523.5	. 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
2		bnj1523.6	. . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))
3		bnj1523.9	. . . . . . . . . . . . 13 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑦 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦))
4		bnj1523.7	. . . . . . . . . . . . . 14 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))
5		bnj1523.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
6		bnj1523.2	. . . . . . . . . . . . . . . . 17 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7		bnj1523.3	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
8		bnj1523.4	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ∪ 𝐶
9	5, 6, 7, 8	bnj60 34629	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
10	1, 9	bnj835 34326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11	2, 10	bnj832 34325	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝐹 Fn 𝐴)
12	4, 11	bnj835 34326	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝐹 Fn 𝐴)
13	3, 12	bnj835 34326	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝐹 Fn 𝐴)
14	1	simp2bi 1144	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn 𝐴)
15	2, 14	bnj832 34325	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝐻 Fn 𝐴)
16	4, 15	bnj835 34326	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝐻 Fn 𝐴)
17	3, 16	bnj835 34326	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝐻 Fn 𝐴)
18		bnj213 34449	. . . . . . . . . . . . 13 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
20	3	simp3bi 1145	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
21	20	bnj1211 34364	. . . . . . . . . . . . . . . 16 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦))
22		con2b 359	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦) ↔ (𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
23	22	albii 1814	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
24	21, 23	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜃 → ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
25		bnj1418 34607	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧𝑅𝑦)
26	25	imim1i 63	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐷))
27	26	alimi 1806	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐷))
28	24, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐷))
29	28	bnj1142 34356	. . . . . . . . . . . . 13 ⊢ (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅) ¬ 𝑧 ∈ 𝐷)
30		bnj1523.8	. . . . . . . . . . . . . 14 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}
31	5	bnj1309 34589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
32	7, 31	bnj1307 34590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
33	32	nfcii 2882	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝐶
34	33	nfuni 4910	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∪ 𝐶
35	8, 34	nfcxfr 2896	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐹
36	35	nfcrii 2890	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
37	30, 36	bnj1534 34420	. . . . . . . . . . . . 13 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
38	29, 18, 37	bnj1533 34419	. . . . . . . . . . . 12 ⊢ (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅)(𝐹‘𝑧) = (𝐻‘𝑧))
39	13, 17, 19, 38	bnj1536 34421	. . . . . . . . . . 11 ⊢ (𝜃 → (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) = (𝐻 ↾ pred(𝑦, 𝐴, 𝑅)))
40	39	opeq2d 4876	. . . . . . . . . 10 ⊢ (𝜃 → ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
41	40	fveq2d 6895	. . . . . . . . 9 ⊢ (𝜃 → (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
42	5, 6, 7, 8	bnj1500 34635	. . . . . . . . . . . . . . 15 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
43	1, 42	bnj835 34326	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
44	2, 43	bnj832 34325	. . . . . . . . . . . . 13 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
45	4, 44	bnj835 34326	. . . . . . . . . . . 12 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
46	45, 36	bnj1529 34637	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
47	3, 46	bnj835 34326	. . . . . . . . . 10 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
48	30	ssrab3 4076	. . . . . . . . . . 11 ⊢ 𝐷 ⊆ 𝐴
49	3	simp2bi 1144	. . . . . . . . . . 11 ⊢ (𝜃 → 𝑦 ∈ 𝐷)
50	48, 49	bnj1213 34365	. . . . . . . . . 10 ⊢ (𝜃 → 𝑦 ∈ 𝐴)
51	47, 50	bnj1294 34384	. . . . . . . . 9 ⊢ (𝜃 → (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
52	1	simp3bi 1145	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
53	2, 52	bnj832 34325	. . . . . . . . . . . . 13 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
54	4, 53	bnj835 34326	. . . . . . . . . . . 12 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55		ax-5 1906	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐻 → ∀𝑥 𝑣 ∈ 𝐻)
56	54, 55	bnj1529 34637	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
57	3, 56	bnj835 34326	. . . . . . . . . 10 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
58	57, 50	bnj1294 34384	. . . . . . . . 9 ⊢ (𝜃 → (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
59	41, 51, 58	3eqtr4d 2777	. . . . . . . 8 ⊢ (𝜃 → (𝐹‘𝑦) = (𝐻‘𝑦))
60	30, 36	bnj1534 34420	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) ≠ (𝐻‘𝑦)}
61	60	bnj1538 34422	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) ≠ (𝐻‘𝑦))
62	3, 61	bnj836 34327	. . . . . . . . 9 ⊢ (𝜃 → (𝐹‘𝑦) ≠ (𝐻‘𝑦))
63	62	neneqd 2940	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝐹‘𝑦) = (𝐻‘𝑦))
64	59, 63	pm2.65i 193	. . . . . . 7 ⊢ ¬ 𝜃
65	64	nex 1795	. . . . . 6 ⊢ ¬ ∃𝑦𝜃
66	1	simp1bi 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
67	2, 66	bnj832 34325	. . . . . . . . 9 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
68	4, 67	bnj835 34326	. . . . . . . 8 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
69	48	a1i 11	. . . . . . . 8 ⊢ (𝜒 → 𝐷 ⊆ 𝐴)
70	4	simp2bi 1144	. . . . . . . . . 10 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
71	4	simp3bi 1145	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹‘𝑥) ≠ (𝐻‘𝑥))
72	30	reqabi 3449	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))
73	70, 71, 72	sylanbrc 582	. . . . . . . . 9 ⊢ (𝜒 → 𝑥 ∈ 𝐷)
74	73	ne0d 4331	. . . . . . . 8 ⊢ (𝜒 → 𝐷 ≠ ∅)
75		bnj69 34577	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅) → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
76	68, 69, 74, 75	syl3anc 1369	. . . . . . 7 ⊢ (𝜒 → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
77	76, 3	bnj1209 34363	. . . . . 6 ⊢ (𝜒 → ∃𝑦𝜃)
78	65, 77	mto 196	. . . . 5 ⊢ ¬ 𝜒
79	78	nex 1795	. . . 4 ⊢ ¬ ∃𝑥𝜒
80	2	simprbi 496	. . . . . 6 ⊢ (𝜓 → 𝐹 ≠ 𝐻)
81	11, 15, 80, 36	bnj1542 34424	. . . . 5 ⊢ (𝜓 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐻‘𝑥))
82	5, 6, 7, 8, 1, 2	bnj1525 34636	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
83	81, 4, 82	bnj1521 34418	. . . 4 ⊢ (𝜓 → ∃𝑥𝜒)
84	79, 83	mto 196	. . 3 ⊢ ¬ 𝜓
85	2, 84	bnj1541 34423	. 2 ⊢ (𝜑 → 𝐹 = 𝐻)
86	1, 85	sylbir 234	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2704   ≠ wne 2935  ∀wral 3056  ∃wrex 3065  {crab 3427   ⊆ wss 3944  ∅c0 4318  ⟨cop 4630  ∪ cuni 4903   class class class wbr 5142   ↾ cres 5674   Fn wfn 6537  ‘cfv 6542   predc-bnj14 34255   FrSe w-bnj15 34259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7865  df-1o 8480  df-bnj17 34254  df-bnj14 34256  df-bnj13 34258  df-bnj15 34260  df-bnj18 34262  df-bnj19 34264

This theorem is referenced by:  bnj1522  34639