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Theorem bnj1448 32433
 Description: Technical lemma for bnj60 32448. 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
bnj1448.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1448.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1448.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1448.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1448.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1448.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1448.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1448.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1448.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1448.10 𝑃 = 𝐻
bnj1448.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1448.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1448.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
Assertion
Ref Expression
bnj1448 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑥,𝑓   𝑧,𝑓
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑧,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,𝑦,𝑧,𝑑)   𝐺(𝑥,𝑦,𝑧,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 bnj1448
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1448.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1448.10 . . . . . . 7 𝑃 = 𝐻
3 bnj1448.9 . . . . . . . . . 10 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
43bnj1317 32207 . . . . . . . . 9 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
54nfcii 2943 . . . . . . . 8 𝑓𝐻
65nfuni 4810 . . . . . . 7 𝑓 𝐻
72, 6nfcxfr 2956 . . . . . 6 𝑓𝑃
8 nfcv 2958 . . . . . . . 8 𝑓𝑥
9 nfcv 2958 . . . . . . . . 9 𝑓𝐺
10 bnj1448.11 . . . . . . . . . 10 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11 nfcv 2958 . . . . . . . . . . . 12 𝑓 pred(𝑥, 𝐴, 𝑅)
127, 11nfres 5824 . . . . . . . . . . 11 𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
138, 12nfop 4784 . . . . . . . . . 10 𝑓𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
1410, 13nfcxfr 2956 . . . . . . . . 9 𝑓𝑍
159, 14nffv 6659 . . . . . . . 8 𝑓(𝐺𝑍)
168, 15nfop 4784 . . . . . . 7 𝑓𝑥, (𝐺𝑍)⟩
1716nfsn 4606 . . . . . 6 𝑓{⟨𝑥, (𝐺𝑍)⟩}
187, 17nfun 4095 . . . . 5 𝑓(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
191, 18nfcxfr 2956 . . . 4 𝑓𝑄
20 nfcv 2958 . . . 4 𝑓𝑧
2119, 20nffv 6659 . . 3 𝑓(𝑄𝑧)
22 bnj1448.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
23 nfcv 2958 . . . . . . 7 𝑓 pred(𝑧, 𝐴, 𝑅)
2419, 23nfres 5824 . . . . . 6 𝑓(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
2520, 24nfop 4784 . . . . 5 𝑓𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
2622, 25nfcxfr 2956 . . . 4 𝑓𝑊
279, 26nffv 6659 . . 3 𝑓(𝐺𝑊)
2821, 27nfeq 2971 . 2 𝑓(𝑄𝑧) = (𝐺𝑊)
2928nf5ri 2194 1 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113  [wsbc 3723   ∪ cun 3882   ⊆ wss 3884  ∅c0 4246  {csn 4528  ⟨cop 4534  ∪ cuni 4803   class class class wbr 5033  dom cdm 5523   ↾ cres 5525   Fn wfn 6323  ‘cfv 6328   predc-bnj14 32072   FrSe w-bnj15 32076   trClc-bnj18 32078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-res 5535  df-iota 6287  df-fv 6336 This theorem is referenced by:  bnj1450  32436  bnj1463  32441
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