Theorem bnj1523 33051

Description: Technical lemma for bnj1522 33052. 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 33042	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
10	1, 9	bnj835 32739	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11	2, 10	bnj832 32738	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝐹 Fn 𝐴)
12	4, 11	bnj835 32739	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝐹 Fn 𝐴)
13	3, 12	bnj835 32739	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝐹 Fn 𝐴)
14	1	simp2bi 1145	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn 𝐴)
15	2, 14	bnj832 32738	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝐻 Fn 𝐴)
16	4, 15	bnj835 32739	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝐻 Fn 𝐴)
17	3, 16	bnj835 32739	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝐻 Fn 𝐴)
18		bnj213 32862	. . . . . . . . . . . . 13 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
20	3	simp3bi 1146	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
21	20	bnj1211 32777	. . . . . . . . . . . . . . . 16 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦))
22		con2b 360	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦) ↔ (𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
23	22	albii 1822	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ 𝐷 → ¬ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
24	21, 23	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜃 → ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧 ∈ 𝐷))
25		bnj1418 33020	. . . . . . . . . . . . . . . . 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 1814	. . . . . . . . . . . . . . 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 32769	. . . . . . . . . . . . 13 ⊢ (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅) ¬ 𝑧 ∈ 𝐷)
30		bnj1523.8	. . . . . . . . . . . . . 14 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}
31	5	bnj1309 33002	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
32	7, 31	bnj1307 33003	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
33	32	nfcii 2891	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝐶
34	33	nfuni 4846	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∪ 𝐶
35	8, 34	nfcxfr 2905	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐹
36	35	nfcrii 2899	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
37	30, 36	bnj1534 32833	. . . . . . . . . . . . 13 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
38	29, 18, 37	bnj1533 32832	. . . . . . . . . . . 12 ⊢ (𝜃 → ∀𝑧 ∈ pred (𝑦, 𝐴, 𝑅)(𝐹‘𝑧) = (𝐻‘𝑧))
39	13, 17, 19, 38	bnj1536 32834	. . . . . . . . . . 11 ⊢ (𝜃 → (𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) = (𝐻 ↾ pred(𝑦, 𝐴, 𝑅)))
40	39	opeq2d 4811	. . . . . . . . . 10 ⊢ (𝜃 → ⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩ = ⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩)
41	40	fveq2d 6778	. . . . . . . . 9 ⊢ (𝜃 → (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
42	5, 6, 7, 8	bnj1500 33048	. . . . . . . . . . . . . . 15 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
43	1, 42	bnj835 32739	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
44	2, 43	bnj832 32738	. . . . . . . . . . . . 13 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
45	4, 44	bnj835 32739	. . . . . . . . . . . 12 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
46	45, 36	bnj1529 33050	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
47	3, 46	bnj835 32739	. . . . . . . . . 10 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
48	30	ssrab3 4015	. . . . . . . . . . 11 ⊢ 𝐷 ⊆ 𝐴
49	3	simp2bi 1145	. . . . . . . . . . 11 ⊢ (𝜃 → 𝑦 ∈ 𝐷)
50	48, 49	bnj1213 32778	. . . . . . . . . 10 ⊢ (𝜃 → 𝑦 ∈ 𝐴)
51	47, 50	bnj1294 32797	. . . . . . . . 9 ⊢ (𝜃 → (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
52	1	simp3bi 1146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
53	2, 52	bnj832 32738	. . . . . . . . . . . . 13 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
54	4, 53	bnj835 32739	. . . . . . . . . . . 12 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55		ax-5 1913	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐻 → ∀𝑥 𝑣 ∈ 𝐻)
56	54, 55	bnj1529 33050	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
57	3, 56	bnj835 32739	. . . . . . . . . 10 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
58	57, 50	bnj1294 32797	. . . . . . . . 9 ⊢ (𝜃 → (𝐻‘𝑦) = (𝐺‘⟨𝑦, (𝐻 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
59	41, 51, 58	3eqtr4d 2788	. . . . . . . 8 ⊢ (𝜃 → (𝐹‘𝑦) = (𝐻‘𝑦))
60	30, 36	bnj1534 32833	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) ≠ (𝐻‘𝑦)}
61	60	bnj1538 32835	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) ≠ (𝐻‘𝑦))
62	3, 61	bnj836 32740	. . . . . . . . 9 ⊢ (𝜃 → (𝐹‘𝑦) ≠ (𝐻‘𝑦))
63	62	neneqd 2948	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝐹‘𝑦) = (𝐻‘𝑦))
64	59, 63	pm2.65i 193	. . . . . . 7 ⊢ ¬ 𝜃
65	64	nex 1803	. . . . . 6 ⊢ ¬ ∃𝑦𝜃
66	1	simp1bi 1144	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
67	2, 66	bnj832 32738	. . . . . . . . 9 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
68	4, 67	bnj835 32739	. . . . . . . 8 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
69	48	a1i 11	. . . . . . . 8 ⊢ (𝜒 → 𝐷 ⊆ 𝐴)
70	4	simp2bi 1145	. . . . . . . . . 10 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
71	4	simp3bi 1146	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹‘𝑥) ≠ (𝐻‘𝑥))
72	30	rabeq2i 3422	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))
73	70, 71, 72	sylanbrc 583	. . . . . . . . 9 ⊢ (𝜒 → 𝑥 ∈ 𝐷)
74	73	ne0d 4269	. . . . . . . 8 ⊢ (𝜒 → 𝐷 ≠ ∅)
75		bnj69 32990	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅) → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
76	68, 69, 74, 75	syl3anc 1370	. . . . . . 7 ⊢ (𝜒 → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦)
77	76, 3	bnj1209 32776	. . . . . 6 ⊢ (𝜒 → ∃𝑦𝜃)
78	65, 77	mto 196	. . . . 5 ⊢ ¬ 𝜒
79	78	nex 1803	. . . 4 ⊢ ¬ ∃𝑥𝜒
80	2	simprbi 497	. . . . . 6 ⊢ (𝜓 → 𝐹 ≠ 𝐻)
81	11, 15, 80, 36	bnj1542 32837	. . . . 5 ⊢ (𝜓 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐻‘𝑥))
82	5, 6, 7, 8, 1, 2	bnj1525 33049	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
83	81, 4, 82	bnj1521 32831	. . . 4 ⊢ (𝜓 → ∃𝑥𝜒)
84	79, 83	mto 196	. . 3 ⊢ ¬ 𝜓
85	2, 84	bnj1541 32836	. 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 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  ∅c0 4256  ⟨cop 4567  ∪ cuni 4839   class class class wbr 5074   ↾ cres 5591   Fn wfn 6428  ‘cfv 6433   predc-bnj14 32667   FrSe w-bnj15 32671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676

This theorem is referenced by:  bnj1522  33052