Theorem bnj1280 35378

Description: Technical lemma for bnj60 35420. 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
bnj1280.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1280.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1280.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1280.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1280.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1280.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1280.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
bnj1280.17	⊢ (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)

Assertion

Ref	Expression
bnj1280	⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑑,𝑓   𝐵,𝑓,𝑔   𝐵,ℎ,𝑓   𝐷,𝑑,𝑥   𝑓,𝐺,𝑔   ℎ,𝐺   𝑅,𝑑,𝑓   𝑔,𝑌   ℎ,𝑌   𝑔,𝑑   𝑥,𝑓,𝑔   ℎ,𝑑,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐴(𝑥,𝑦,𝑔,ℎ)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐷(𝑦,𝑓,𝑔,ℎ)   𝑅(𝑥,𝑦,𝑔,ℎ)   𝐸(𝑥,𝑦,𝑓,𝑔,ℎ,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1280

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1280.1	. . . . . . . 8 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1280.2	. . . . . . . 8 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1280.3	. . . . . . . 8 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		bnj1280.4	. . . . . . . 8 ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
5		bnj1280.5	. . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
6		bnj1280.6	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
7		bnj1280.7	. . . . . . . 8 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
8	1, 2, 3, 4, 5, 6, 7	bnj1286 35377	. . . . . . 7 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
9	8	sseld 3944	. . . . . 6 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧 ∈ 𝐷))
10		bnj1280.17	. . . . . . . . 9 ⊢ (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
11		disj1 4418	. . . . . . . . 9 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅ ↔ ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐸))
12	10, 11	sylib 221	. . . . . . . 8 ⊢ (𝜓 → ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐸))
13	12	19.21bi 2232	. . . . . . 7 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐸))
14		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
15		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (ℎ‘𝑥) = (ℎ‘𝑧))
16	14, 15	neeq12d 3026	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑔‘𝑥) ≠ (ℎ‘𝑥) ↔ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
17	16, 5	elrab2 3662	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐸 ↔ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
18	17	notbii 323	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝐸 ↔ ¬ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
19		imnan 404	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐷 → ¬ (𝑔‘𝑧) ≠ (ℎ‘𝑧)) ↔ ¬ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
20		nne 2969	. . . . . . . . 9 ⊢ (¬ (𝑔‘𝑧) ≠ (ℎ‘𝑧) ↔ (𝑔‘𝑧) = (ℎ‘𝑧))
21	20	imbi2i 339	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐷 → ¬ (𝑔‘𝑧) ≠ (ℎ‘𝑧)) ↔ (𝑧 ∈ 𝐷 → (𝑔‘𝑧) = (ℎ‘𝑧)))
22	18, 19, 21	3bitr2i 302	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝐸 ↔ (𝑧 ∈ 𝐷 → (𝑔‘𝑧) = (ℎ‘𝑧)))
23	13, 22	imbitrdi 254	. . . . . 6 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑧 ∈ 𝐷 → (𝑔‘𝑧) = (ℎ‘𝑧))))
24	9, 23	mpdd 44	. . . . 5 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑔‘𝑧) = (ℎ‘𝑧)))
25	24	imp 411	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑔‘𝑧) = (ℎ‘𝑧))
26		fvres 6904	. . . . . 6 ⊢ (𝑧 ∈ 𝐷 → ((𝑔 ↾ 𝐷)‘𝑧) = (𝑔‘𝑧))
27	9, 26	syl6 36	. . . . 5 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((𝑔 ↾ 𝐷)‘𝑧) = (𝑔‘𝑧)))
28	27	imp 411	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔 ↾ 𝐷)‘𝑧) = (𝑔‘𝑧))
29		fvres 6904	. . . . . 6 ⊢ (𝑧 ∈ 𝐷 → ((ℎ ↾ 𝐷)‘𝑧) = (ℎ‘𝑧))
30	9, 29	syl6 36	. . . . 5 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((ℎ ↾ 𝐷)‘𝑧) = (ℎ‘𝑧)))
31	30	imp 411	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((ℎ ↾ 𝐷)‘𝑧) = (ℎ‘𝑧))
32	25, 28, 31	3eqtr4d 2815	. . 3 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧))
33	32	ralrimiva 3164	. 2 ⊢ (𝜓 → ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧))
34	8	resabs1d 6011	. . . 4 ⊢ (𝜓 → ((𝑔 ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
35	8	resabs1d 6011	. . . 4 ⊢ (𝜓 → ((ℎ ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
36	34, 35	eqeq12d 2786	. . 3 ⊢ (𝜓 → (((𝑔 ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((ℎ ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅))))
37	1, 2, 3, 4, 5, 6, 7	bnj1256 35373	. . . . . . 7 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
38	4	bnj1292 35173	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom 𝑔
39		fndm 6642	. . . . . . . . 9 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
40	38, 39	sseqtrid 3987	. . . . . . . 8 ⊢ (𝑔 Fn 𝑑 → 𝐷 ⊆ 𝑑)
41		fnssres 6662	. . . . . . . 8 ⊢ ((𝑔 Fn 𝑑 ∧ 𝐷 ⊆ 𝑑) → (𝑔 ↾ 𝐷) Fn 𝐷)
42	40, 41	mpdan 699	. . . . . . 7 ⊢ (𝑔 Fn 𝑑 → (𝑔 ↾ 𝐷) Fn 𝐷)
43	37, 42	bnj31 35078	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 (𝑔 ↾ 𝐷) Fn 𝐷)
44	43	bnj1265 35170	. . . . 5 ⊢ (𝜑 → (𝑔 ↾ 𝐷) Fn 𝐷)
45	7, 44	bnj835 35118	. . . 4 ⊢ (𝜓 → (𝑔 ↾ 𝐷) Fn 𝐷)
46	1, 2, 3, 4, 5, 6, 7	bnj1259 35374	. . . . . . 7 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
47	4	bnj1293 35174	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom ℎ
48		fndm 6642	. . . . . . . . 9 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
49	47, 48	sseqtrid 3987	. . . . . . . 8 ⊢ (ℎ Fn 𝑑 → 𝐷 ⊆ 𝑑)
50		fnssres 6662	. . . . . . . 8 ⊢ ((ℎ Fn 𝑑 ∧ 𝐷 ⊆ 𝑑) → (ℎ ↾ 𝐷) Fn 𝐷)
51	49, 50	mpdan 699	. . . . . . 7 ⊢ (ℎ Fn 𝑑 → (ℎ ↾ 𝐷) Fn 𝐷)
52	46, 51	bnj31 35078	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 (ℎ ↾ 𝐷) Fn 𝐷)
53	52	bnj1265 35170	. . . . 5 ⊢ (𝜑 → (ℎ ↾ 𝐷) Fn 𝐷)
54	7, 53	bnj835 35118	. . . 4 ⊢ (𝜓 → (ℎ ↾ 𝐷) Fn 𝐷)
55		fvreseq 7039	. . . 4 ⊢ ((((𝑔 ↾ 𝐷) Fn 𝐷 ∧ (ℎ ↾ 𝐷) Fn 𝐷) ∧ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷) → (((𝑔 ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((ℎ ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧)))
56	45, 54, 8, 55	syl21anc 850	. . 3 ⊢ (𝜓 → (((𝑔 ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((ℎ ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧)))
57	36, 56	bitr3d 284	. 2 ⊢ (𝜓 → ((𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧)))
58	33, 57	mpbird 260	1 ⊢ (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1566   = wceq 1568   ∈ wcel 2150  {cab 2748   ≠ wne 2965  ∀wral 3086  ∃wrex 3096  {crab 3423   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ⟨cop 4600   class class class wbr 5114  dom cdm 5665   ↾ cres 5667   Fn wfn 6535  ‘cfv 6540   ∧ w-bnj17 35045   predc-bnj14 35047   FrSe w-bnj15 35051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-bnj17 35046

This theorem is referenced by:  bnj1311  35382