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Theorem bnj1525 32238
Description: Technical lemma for bnj1522 32241. 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
bnj1525.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1525.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1525.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1525.4 𝐹 = 𝐶
bnj1525.5 (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
bnj1525.6 (𝜓 ↔ (𝜑𝐹𝐻))
Assertion
Ref Expression
bnj1525 (𝜓 → ∀𝑥𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐻   𝑥,𝑅   𝑥,𝑑   𝑥,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑑)   𝜓(𝑥,𝑓,𝑑)   𝐴(𝑓,𝑑)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑓,𝑑)   𝐻(𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1525
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1525.6 . . 3 (𝜓 ↔ (𝜑𝐹𝐻))
2 bnj1525.5 . . . . 5 (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
3 nfv 1906 . . . . . 6 𝑥 𝑅 FrSe 𝐴
4 nfv 1906 . . . . . 6 𝑥 𝐻 Fn 𝐴
5 nfra1 3216 . . . . . 6 𝑥𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
63, 4, 5nf3an 1893 . . . . 5 𝑥(𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
72, 6nfxfr 1844 . . . 4 𝑥𝜑
8 bnj1525.4 . . . . . 6 𝐹 = 𝐶
9 bnj1525.3 . . . . . . . . 9 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
10 bnj1525.1 . . . . . . . . . 10 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
1110bnj1309 32191 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
129, 11bnj1307 32192 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
1312nfcii 2962 . . . . . . 7 𝑥𝐶
1413nfuni 4837 . . . . . 6 𝑥 𝐶
158, 14nfcxfr 2972 . . . . 5 𝑥𝐹
16 nfcv 2974 . . . . 5 𝑥𝐻
1715, 16nfne 3116 . . . 4 𝑥 𝐹𝐻
187, 17nfan 1891 . . 3 𝑥(𝜑𝐹𝐻)
191, 18nfxfr 1844 . 2 𝑥𝜓
2019nf5ri 2185 1 (𝜓 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wal 1526   = wceq 1528  {cab 2796  wne 3013  wral 3135  wrex 3136  wss 3933  cop 4563   cuni 4830  cres 5550   Fn wfn 6343  cfv 6348   predc-bnj14 31857   FrSe w-bnj15 31861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-uni 4831
This theorem is referenced by:  bnj1523  32240
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