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Theorem bnj1446 31412
Description: Technical lemma for bnj60 31429. 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 2894 . . . . . . . . . . 11 𝑑 pred(𝑥, 𝐴, 𝑅)
5 bnj1446.8 . . . . . . . . . . . 12 (𝜏′[𝑦 / 𝑥]𝜏)
6 nfcv 2894 . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . . . . . 18 𝑑𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
109nfab 2899 . . . . . . . . . . . . . . . . 17 𝑑{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
118, 10nfcxfr 2892 . . . . . . . . . . . . . . . 16 𝑑𝐶
1211nfcri 2888 . . . . . . . . . . . . . . 15 𝑑 𝑓𝐶
13 nfv 1984 . . . . . . . . . . . . . . 15 𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
1412, 13nfan 1969 . . . . . . . . . . . . . 14 𝑑(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
157, 14nfxfr 1920 . . . . . . . . . . . . 13 𝑑𝜏
166, 15nfsbc 3590 . . . . . . . . . . . 12 𝑑[𝑦 / 𝑥]𝜏
175, 16nfxfr 1920 . . . . . . . . . . 11 𝑑𝜏′
184, 17nfrex 3137 . . . . . . . . . 10 𝑑𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
1918nfab 2899 . . . . . . . . 9 𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
203, 19nfcxfr 2892 . . . . . . . 8 𝑑𝐻
2120nfuni 4586 . . . . . . 7 𝑑 𝐻
222, 21nfcxfr 2892 . . . . . 6 𝑑𝑃
23 nfcv 2894 . . . . . . . 8 𝑑𝑥
24 nfcv 2894 . . . . . . . . 9 𝑑𝐺
25 bnj1446.11 . . . . . . . . . 10 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2622, 4nfres 5545 . . . . . . . . . . 11 𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
2723, 26nfop 4561 . . . . . . . . . 10 𝑑𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
2825, 27nfcxfr 2892 . . . . . . . . 9 𝑑𝑍
2924, 28nffv 6351 . . . . . . . 8 𝑑(𝐺𝑍)
3023, 29nfop 4561 . . . . . . 7 𝑑𝑥, (𝐺𝑍)⟩
3130nfsn 4378 . . . . . 6 𝑑{⟨𝑥, (𝐺𝑍)⟩}
3222, 31nfun 3904 . . . . 5 𝑑(𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
331, 32nfcxfr 2892 . . . 4 𝑑𝑄
34 nfcv 2894 . . . 4 𝑑𝑧
3533, 34nffv 6351 . . 3 𝑑(𝑄𝑧)
36 bnj1446.13 . . . . 5 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
37 nfcv 2894 . . . . . . 7 𝑑 pred(𝑧, 𝐴, 𝑅)
3833, 37nfres 5545 . . . . . 6 𝑑(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
3934, 38nfop 4561 . . . . 5 𝑑𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
4036, 39nfcxfr 2892 . . . 4 𝑑𝑊
4124, 40nffv 6351 . . 3 𝑑(𝐺𝑊)
4235, 41nfeq 2906 . 2 𝑑(𝑄𝑧) = (𝐺𝑊)
4342nf5ri 2204 1 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072  wal 1622   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wne 2924  wral 3042  wrex 3043  {crab 3046  [wsbc 3568  cun 3705  wss 3707  c0 4050  {csn 4313  cop 4319   cuni 4580   class class class wbr 4796  dom cdm 5258  cres 5260   Fn wfn 6036  cfv 6041   predc-bnj14 31055   FrSe w-bnj15 31059   trClc-bnj18 31061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-xp 5264  df-res 5270  df-iota 6004  df-fv 6049
This theorem is referenced by:  bnj1450  31417  bnj1463  31422
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