Theorem bnj1280 35570

Description: Technical lemma for bnj60 35612. 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 35569	. . . . . . 7 ⊢ (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
9	8	sseld 3930	. . . . . 6 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧 ∈ 𝐷))
10		bnj1280.17	. . . . . . . . 9 ⊢ (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
11		disj1 4405	. . . . . . . . 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 2225	. . . . . . 7 ⊢ (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧 ∈ 𝐸))
14		fveq2 6874	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
15		fveq2 6874	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (ℎ‘𝑥) = (ℎ‘𝑧))
16	14, 15	neeq12d 3016	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑔‘𝑥) ≠ (ℎ‘𝑥) ↔ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
17	16, 5	elrab2 3649	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐸 ↔ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
18	17	notbii 323	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝐸 ↔ ¬ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
19		imnan 405	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐷 → ¬ (𝑔‘𝑧) ≠ (ℎ‘𝑧)) ↔ ¬ (𝑧 ∈ 𝐷 ∧ (𝑔‘𝑧) ≠ (ℎ‘𝑧)))
20		nne 2959	. . . . . . . . 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 412	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑔‘𝑧) = (ℎ‘𝑧))
26		fvres 6893	. . . . . 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 412	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔 ↾ 𝐷)‘𝑧) = (𝑔‘𝑧))
29		fvres 6893	. . . . . 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 412	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((ℎ ↾ 𝐷)‘𝑧) = (ℎ‘𝑧))
32	25, 28, 31	3eqtr4d 2805	. . 3 ⊢ ((𝜓 ∧ 𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧))
33	32	ralrimiva 3154	. 2 ⊢ (𝜓 → ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔 ↾ 𝐷)‘𝑧) = ((ℎ ↾ 𝐷)‘𝑧))
34	8	resabs1d 5996	. . . 4 ⊢ (𝜓 → ((𝑔 ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
35	8	resabs1d 5996	. . . 4 ⊢ (𝜓 → ((ℎ ↾ 𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (ℎ ↾ pred(𝑥, 𝐴, 𝑅)))
36	34, 35	eqeq12d 2776	. . 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 35565	. . . . . . 7 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
38	4	bnj1292 35365	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom 𝑔
39		fndm 6631	. . . . . . . . 9 ⊢ (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
40	38, 39	sseqtrid 3973	. . . . . . . 8 ⊢ (𝑔 Fn 𝑑 → 𝐷 ⊆ 𝑑)
41		fnssres 6651	. . . . . . . 8 ⊢ ((𝑔 Fn 𝑑 ∧ 𝐷 ⊆ 𝑑) → (𝑔 ↾ 𝐷) Fn 𝐷)
42	40, 41	mpdan 700	. . . . . . 7 ⊢ (𝑔 Fn 𝑑 → (𝑔 ↾ 𝐷) Fn 𝐷)
43	37, 42	bnj31 35270	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 (𝑔 ↾ 𝐷) Fn 𝐷)
44	43	bnj1265 35362	. . . . 5 ⊢ (𝜑 → (𝑔 ↾ 𝐷) Fn 𝐷)
45	7, 44	bnj835 35310	. . . 4 ⊢ (𝜓 → (𝑔 ↾ 𝐷) Fn 𝐷)
46	1, 2, 3, 4, 5, 6, 7	bnj1259 35566	. . . . . . 7 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
47	4	bnj1293 35366	. . . . . . . . 9 ⊢ 𝐷 ⊆ dom ℎ
48		fndm 6631	. . . . . . . . 9 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
49	47, 48	sseqtrid 3973	. . . . . . . 8 ⊢ (ℎ Fn 𝑑 → 𝐷 ⊆ 𝑑)
50		fnssres 6651	. . . . . . . 8 ⊢ ((ℎ Fn 𝑑 ∧ 𝐷 ⊆ 𝑑) → (ℎ ↾ 𝐷) Fn 𝐷)
51	49, 50	mpdan 700	. . . . . . 7 ⊢ (ℎ Fn 𝑑 → (ℎ ↾ 𝐷) Fn 𝐷)
52	46, 51	bnj31 35270	. . . . . 6 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 (ℎ ↾ 𝐷) Fn 𝐷)
53	52	bnj1265 35362	. . . . 5 ⊢ (𝜑 → (ℎ ↾ 𝐷) Fn 𝐷)
54	7, 53	bnj835 35310	. . . 4 ⊢ (𝜓 → (ℎ ↾ 𝐷) Fn 𝐷)
55		fvreseq 7028	. . . 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 851	. . 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   class class class wbr 5103  dom cdm 5648   ↾ cres 5650   Fn wfn 6523  ‘cfv 6528   ∧ w-bnj17 35237   predc-bnj14 35239   FrSe w-bnj15 35243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-bnj17 35238

This theorem is used by:  bnj1311  35574