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Theorem bnj1446 31087
Description: Technical lemma for bnj60 31104. 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
bnj1446.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1446.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1446.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1446.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1446.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1446.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1446.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1446.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1446.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1446.10 𝑃 = 𝐻
bnj1446.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1446.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1446.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
Assertion
Ref Expression
bnj1446 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
Distinct variable groups:   𝐴,𝑑,𝑥   𝐵,𝑓   𝐺,𝑑   𝑅,𝑑,𝑥   𝑓,𝑑,𝑥   𝑦,𝑑,𝑥   𝑧,𝑑
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑧,𝑓)   𝐵(𝑥,𝑦,𝑧,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(𝑦,𝑧,𝑓)   𝐺(𝑥,𝑦,𝑧,𝑓)   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 bnj1446
StepHypRef Expression
1 bnj1446.12 . . . . 5 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1446.10 . . . . . . 7 𝑃 = 𝐻
3 bnj1446.9 . . . . . . . . 9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4 nfcv 2762 . . . . . . . . . . 11 𝑑 pred(𝑥, 𝐴, 𝑅)
5 bnj1446.8 . . . . . . . . . . . 12 (𝜏′[𝑦 / 𝑥]𝜏)
6 nfcv 2762 . . . . . . . . . . . . 13 𝑑𝑦
7 bnj1446.4 . . . . . . . . . . . . . 14 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
8 bnj1446.3 . . . . . . . . . . . . . . . . 17 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
9 nfre1 3002 . . . . . . . . . . . . . . . . . 18 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
109nfab 2766 . . . . . . . . . . . . . . . . 17 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
118, 10nfcxfr 2760 . . . . . . . . . . . . . . . 16 𝑑𝐶
1211nfcri 2756 . . . . . . . . . . . . . . 15 𝑑 𝑓𝐶
13 nfv 1841 . . . . . . . . . . . . . . 15 𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
1412, 13nfan 1826 . . . . . . . . . . . . . 14 𝑑(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
157, 14nfxfr 1777 . . . . . . . . . . . . 13 𝑑𝜏
166, 15nfsbc 3451 . . . . . . . . . . . 12 𝑑[𝑦 / 𝑥]𝜏
175, 16nfxfr 1777 . . . . . . . . . . 11 𝑑𝜏′
184, 17nfrex 3004 . . . . . . . . . 10 𝑑𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1918nfab 2766 . . . . . . . . 9 𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
203, 19nfcxfr 2760 . . . . . . . 8 𝑑𝐻
2120nfuni 4433 . . . . . . 7 𝑑 𝐻
222, 21nfcxfr 2760 . . . . . 6 𝑑𝑃
23 nfcv 2762 . . . . . . . 8 𝑑𝑥
24 nfcv 2762 . . . . . . . . 9 𝑑𝐺
25 bnj1446.11 . . . . . . . . . 10 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2622, 4nfres 5387 . . . . . . . . . . 11 𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
2723, 26nfop 4409 . . . . . . . . . 10 𝑑𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2825, 27nfcxfr 2760 . . . . . . . . 9 𝑑𝑍
2924, 28nffv 6185 . . . . . . . 8 𝑑(𝐺𝑍)
3023, 29nfop 4409 . . . . . . 7 𝑑𝑥, (𝐺𝑍)⟩
3130nfsn 4233 . . . . . 6 𝑑{⟨𝑥, (𝐺𝑍)⟩}
3222, 31nfun 3761 . . . . 5 𝑑(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
331, 32nfcxfr 2760 . . . 4 𝑑𝑄
34 nfcv 2762 . . . 4 𝑑𝑧
3533, 34nffv 6185 . . 3 𝑑(𝑄𝑧)
36 bnj1446.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
37 nfcv 2762 . . . . . . 7 𝑑 pred(𝑧, 𝐴, 𝑅)
3833, 37nfres 5387 . . . . . 6 𝑑(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
3934, 38nfop 4409 . . . . 5 𝑑𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
4036, 39nfcxfr 2760 . . . 4 𝑑𝑊
4124, 40nffv 6185 . . 3 𝑑(𝐺𝑊)
4235, 41nfeq 2773 . 2 𝑑(𝑄𝑧) = (𝐺𝑊)
4342nf5ri 2063 1 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  {crab 2913  [wsbc 3429  cun 3565  wss 3567  c0 3907  {csn 4168  cop 4174   cuni 4427   class class class wbr 4644  dom cdm 5104  cres 5106   Fn wfn 5871  cfv 5876   predc-bnj14 30728   FrSe w-bnj15 30732   trClc-bnj18 30734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-res 5116  df-iota 5839  df-fv 5884
This theorem is referenced by:  bnj1450  31092  bnj1463  31097
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