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Theorem bnj1280 31420
Description: Technical lemma for bnj60 31462. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1280.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1280.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1280.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1280.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1280.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1280.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
bnj1280.17 (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
Assertion
Ref Expression
bnj1280 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑑,𝑓   𝐵,𝑓,𝑔   𝐵,,𝑓   𝐷,𝑑,𝑥   𝑓,𝐺,𝑔   ,𝐺   𝑅,𝑑,𝑓   𝑔,𝑌   ,𝑌   𝑔,𝑑   𝑥,𝑓,𝑔   ,𝑑,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑔,)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,)   𝑅(𝑥,𝑦,𝑔,)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1280
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj1280.1 . . . . . . . 8 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1280.2 . . . . . . . 8 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1280.3 . . . . . . . 8 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1280.4 . . . . . . . 8 𝐷 = (dom 𝑔 ∩ dom )
5 bnj1280.5 . . . . . . . 8 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
6 bnj1280.6 . . . . . . . 8 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
7 bnj1280.7 . . . . . . . 8 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
81, 2, 3, 4, 5, 6, 7bnj1286 31419 . . . . . . 7 (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
98sseld 3808 . . . . . 6 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝐷))
10 bnj1280.17 . . . . . . . . 9 (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
11 disj1 4227 . . . . . . . . 9 (( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅ ↔ ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
1210, 11sylib 209 . . . . . . . 8 (𝜓 → ∀𝑧(𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
131219.21bi 2225 . . . . . . 7 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑧𝐸))
14 fveq2 6415 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑔𝑥) = (𝑔𝑧))
15 fveq2 6415 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥) = (𝑧))
1614, 15neeq12d 3050 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑔𝑥) ≠ (𝑥) ↔ (𝑔𝑧) ≠ (𝑧)))
1716, 5elrab2 3573 . . . . . . . . 9 (𝑧𝐸 ↔ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
1817notbii 311 . . . . . . . 8 𝑧𝐸 ↔ ¬ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
19 imnan 388 . . . . . . . 8 ((𝑧𝐷 → ¬ (𝑔𝑧) ≠ (𝑧)) ↔ ¬ (𝑧𝐷 ∧ (𝑔𝑧) ≠ (𝑧)))
20 nne 2993 . . . . . . . . 9 (¬ (𝑔𝑧) ≠ (𝑧) ↔ (𝑔𝑧) = (𝑧))
2120imbi2i 327 . . . . . . . 8 ((𝑧𝐷 → ¬ (𝑔𝑧) ≠ (𝑧)) ↔ (𝑧𝐷 → (𝑔𝑧) = (𝑧)))
2218, 19, 213bitr2i 290 . . . . . . 7 𝑧𝐸 ↔ (𝑧𝐷 → (𝑔𝑧) = (𝑧)))
2313, 22syl6ib 242 . . . . . 6 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑧𝐷 → (𝑔𝑧) = (𝑧))))
249, 23mpdd 43 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑔𝑧) = (𝑧)))
2524imp 395 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑔𝑧) = (𝑧))
26 fvres 6434 . . . . . 6 (𝑧𝐷 → ((𝑔𝐷)‘𝑧) = (𝑔𝑧))
279, 26syl6 35 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((𝑔𝐷)‘𝑧) = (𝑔𝑧)))
2827imp 395 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔𝐷)‘𝑧) = (𝑔𝑧))
29 fvres 6434 . . . . . 6 (𝑧𝐷 → ((𝐷)‘𝑧) = (𝑧))
309, 29syl6 35 . . . . 5 (𝜓 → (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → ((𝐷)‘𝑧) = (𝑧)))
3130imp 395 . . . 4 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝐷)‘𝑧) = (𝑧))
3225, 28, 313eqtr4d 2861 . . 3 ((𝜓𝑧 ∈ pred(𝑥, 𝐴, 𝑅)) → ((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧))
3332ralrimiva 3165 . 2 (𝜓 → ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧))
348resabs1d 5642 . . . 4 (𝜓 → ((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
358resabs1d 5642 . . . 4 (𝜓 → ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
3634, 35eqeq12d 2832 . . 3 (𝜓 → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅))))
371, 2, 3, 4, 5, 6, 7bnj1256 31415 . . . . . . 7 (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
384bnj1292 31218 . . . . . . . . 9 𝐷 ⊆ dom 𝑔
39 fndm 6208 . . . . . . . . 9 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
4038, 39syl5sseq 3861 . . . . . . . 8 (𝑔 Fn 𝑑𝐷𝑑)
41 fnssres 6222 . . . . . . . 8 ((𝑔 Fn 𝑑𝐷𝑑) → (𝑔𝐷) Fn 𝐷)
4240, 41mpdan 670 . . . . . . 7 (𝑔 Fn 𝑑 → (𝑔𝐷) Fn 𝐷)
4337, 42bnj31 31120 . . . . . 6 (𝜑 → ∃𝑑𝐵 (𝑔𝐷) Fn 𝐷)
4443bnj1265 31215 . . . . 5 (𝜑 → (𝑔𝐷) Fn 𝐷)
457, 44bnj835 31161 . . . 4 (𝜓 → (𝑔𝐷) Fn 𝐷)
461, 2, 3, 4, 5, 6, 7bnj1259 31416 . . . . . . 7 (𝜑 → ∃𝑑𝐵 Fn 𝑑)
474bnj1293 31219 . . . . . . . . 9 𝐷 ⊆ dom
48 fndm 6208 . . . . . . . . 9 ( Fn 𝑑 → dom = 𝑑)
4947, 48syl5sseq 3861 . . . . . . . 8 ( Fn 𝑑𝐷𝑑)
50 fnssres 6222 . . . . . . . 8 (( Fn 𝑑𝐷𝑑) → (𝐷) Fn 𝐷)
5149, 50mpdan 670 . . . . . . 7 ( Fn 𝑑 → (𝐷) Fn 𝐷)
5246, 51bnj31 31120 . . . . . 6 (𝜑 → ∃𝑑𝐵 (𝐷) Fn 𝐷)
5352bnj1265 31215 . . . . 5 (𝜑 → (𝐷) Fn 𝐷)
547, 53bnj835 31161 . . . 4 (𝜓 → (𝐷) Fn 𝐷)
55 fvreseq 6548 . . . 4 ((((𝑔𝐷) Fn 𝐷 ∧ (𝐷) Fn 𝐷) ∧ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷) → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5645, 54, 8, 55syl21anc 857 . . 3 (𝜓 → (((𝑔𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) = ((𝐷) ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5736, 56bitr3d 272 . 2 (𝜓 → ((𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ pred (𝑥, 𝐴, 𝑅)((𝑔𝐷)‘𝑧) = ((𝐷)‘𝑧)))
5833, 57mpbird 248 1 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2157  {cab 2803  wne 2989  wral 3107  wrex 3108  {crab 3111  cin 3779  wss 3780  c0 4127  cop 4387   class class class wbr 4855  dom cdm 5322  cres 5324   Fn wfn 6103  cfv 6108  w-bnj17 31087   predc-bnj14 31089   FrSe w-bnj15 31093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-iota 6071  df-fun 6110  df-fn 6111  df-fv 6116  df-bnj17 31088
This theorem is referenced by:  bnj1311  31424
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