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Theorem bnj1446 35037
Description: Technical lemma for bnj60 35054. 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 2902 . . . . . . . . . . 11 𝑑 pred(𝑥, 𝐴, 𝑅)
5 bnj1446.8 . . . . . . . . . . . 12 (𝜏′[𝑦 / 𝑥]𝜏)
6 nfcv 2902 . . . . . . . . . . . . 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 3282 . . . . . . . . . . . . . . . . . 18 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
109nfab 2908 . . . . . . . . . . . . . . . . 17 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
118, 10nfcxfr 2900 . . . . . . . . . . . . . . . 16 𝑑𝐶
1211nfcri 2894 . . . . . . . . . . . . . . 15 𝑑 𝑓𝐶
13 nfv 1911 . . . . . . . . . . . . . . 15 𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
1412, 13nfan 1896 . . . . . . . . . . . . . 14 𝑑(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
157, 14nfxfr 1849 . . . . . . . . . . . . 13 𝑑𝜏
166, 15nfsbcw 3812 . . . . . . . . . . . 12 𝑑[𝑦 / 𝑥]𝜏
175, 16nfxfr 1849 . . . . . . . . . . 11 𝑑𝜏′
184, 17nfrexw 3310 . . . . . . . . . 10 𝑑𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1918nfab 2908 . . . . . . . . 9 𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
203, 19nfcxfr 2900 . . . . . . . 8 𝑑𝐻
2120nfuni 4918 . . . . . . 7 𝑑 𝐻
222, 21nfcxfr 2900 . . . . . 6 𝑑𝑃
23 nfcv 2902 . . . . . . . 8 𝑑𝑥
24 nfcv 2902 . . . . . . . . 9 𝑑𝐺
25 bnj1446.11 . . . . . . . . . 10 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2622, 4nfres 6001 . . . . . . . . . . 11 𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
2723, 26nfop 4893 . . . . . . . . . 10 𝑑𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2825, 27nfcxfr 2900 . . . . . . . . 9 𝑑𝑍
2924, 28nffv 6916 . . . . . . . 8 𝑑(𝐺𝑍)
3023, 29nfop 4893 . . . . . . 7 𝑑𝑥, (𝐺𝑍)⟩
3130nfsn 4711 . . . . . 6 𝑑{⟨𝑥, (𝐺𝑍)⟩}
3222, 31nfun 4179 . . . . 5 𝑑(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
331, 32nfcxfr 2900 . . . 4 𝑑𝑄
34 nfcv 2902 . . . 4 𝑑𝑧
3533, 34nffv 6916 . . 3 𝑑(𝑄𝑧)
36 bnj1446.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
37 nfcv 2902 . . . . . . 7 𝑑 pred(𝑧, 𝐴, 𝑅)
3833, 37nfres 6001 . . . . . 6 𝑑(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
3934, 38nfop 4893 . . . . 5 𝑑𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
4036, 39nfcxfr 2900 . . . 4 𝑑𝑊
4124, 40nffv 6916 . . 3 𝑑(𝐺𝑊)
4235, 41nfeq 2916 . 2 𝑑(𝑄𝑧) = (𝐺𝑊)
4342nf5ri 2192 1 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1534   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  {crab 3432  [wsbc 3790  cun 3960  wss 3962  c0 4338  {csn 4630  cop 4636   cuni 4911   class class class wbr 5147  dom cdm 5688  cres 5690   Fn wfn 6557  cfv 6562   predc-bnj14 34680   FrSe w-bnj15 34684   trClc-bnj18 34686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-res 5700  df-iota 6515  df-fv 6570
This theorem is referenced by:  bnj1450  35042  bnj1463  35047
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