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Theorem bnj1447 32221
 Description: Technical lemma for bnj60 32237. 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
bnj1447.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1447.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1447.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1447.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1447.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1447.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1447.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1447.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1447.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1447.10 𝑃 = 𝐻
bnj1447.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1447.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1447.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
Assertion
Ref Expression
bnj1447 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑅   𝑥,𝑦   𝑦,𝑧   𝑦,𝑓
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑧,𝑓,𝑑)   𝐺(𝑥,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1447
StepHypRef Expression
1 bnj1447.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1447.10 . . . . . . 7 𝑃 = 𝐻
3 bnj1447.9 . . . . . . . . 9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4 nfre1 3311 . . . . . . . . . 10 𝑦𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
54nfab 2989 . . . . . . . . 9 𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
63, 5nfcxfr 2980 . . . . . . . 8 𝑦𝐻
76nfuni 4844 . . . . . . 7 𝑦 𝐻
82, 7nfcxfr 2980 . . . . . 6 𝑦𝑃
9 nfcv 2982 . . . . . . . 8 𝑦𝑥
10 nfcv 2982 . . . . . . . . 9 𝑦𝐺
11 bnj1447.11 . . . . . . . . . 10 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
12 nfcv 2982 . . . . . . . . . . . 12 𝑦 pred(𝑥, 𝐴, 𝑅)
138, 12nfres 5854 . . . . . . . . . . 11 𝑦(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
149, 13nfop 4818 . . . . . . . . . 10 𝑦𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1511, 14nfcxfr 2980 . . . . . . . . 9 𝑦𝑍
1610, 15nffv 6679 . . . . . . . 8 𝑦(𝐺𝑍)
179, 16nfop 4818 . . . . . . 7 𝑦𝑥, (𝐺𝑍)⟩
1817nfsn 4642 . . . . . 6 𝑦{⟨𝑥, (𝐺𝑍)⟩}
198, 18nfun 4145 . . . . 5 𝑦(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
201, 19nfcxfr 2980 . . . 4 𝑦𝑄
21 nfcv 2982 . . . 4 𝑦𝑧
2220, 21nffv 6679 . . 3 𝑦(𝑄𝑧)
23 bnj1447.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
24 nfcv 2982 . . . . . . 7 𝑦 pred(𝑧, 𝐴, 𝑅)
2520, 24nfres 5854 . . . . . 6 𝑦(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
2621, 25nfop 4818 . . . . 5 𝑦𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
2723, 26nfcxfr 2980 . . . 4 𝑦𝑊
2810, 27nffv 6679 . . 3 𝑦(𝐺𝑊)
2922, 28nfeq 2996 . 2 𝑦(𝑄𝑧) = (𝐺𝑊)
3029nf5ri 2187 1 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  {crab 3147  [wsbc 3776   ∪ cun 3938   ⊆ wss 3940  ∅c0 4295  {csn 4564  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063  dom cdm 5554   ↾ cres 5556   Fn wfn 6349  ‘cfv 6354   predc-bnj14 31863   FrSe w-bnj15 31867   trClc-bnj18 31869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-res 5566  df-iota 6313  df-fv 6362 This theorem is referenced by:  bnj1450  32225
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