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Theorem bnj1525 32329
Description: Technical lemma for bnj1522 32332. 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 1908 . . . . . 6 𝑥 𝑅 FrSe 𝐴
4 nfv 1908 . . . . . 6 𝑥 𝐻 Fn 𝐴
5 nfra1 3217 . . . . . 6 𝑥𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
63, 4, 5nf3an 1895 . . . . 5 𝑥(𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
72, 6nfxfr 1846 . . . 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 32282 . . . . . . . . 9 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
129, 11bnj1307 32283 . . . . . . . 8 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
1312nfcii 2963 . . . . . . 7 𝑥𝐶
1413nfuni 4837 . . . . . 6 𝑥 𝐶
158, 14nfcxfr 2973 . . . . 5 𝑥𝐹
16 nfcv 2975 . . . . 5 𝑥𝐻
1715, 16nfne 3117 . . . 4 𝑥 𝐹𝐻
187, 17nfan 1893 . . 3 𝑥(𝜑𝐹𝐻)
191, 18nfxfr 1846 . 2 𝑥𝜓
2019nf5ri 2187 1 (𝜓 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081  wal 1528   = wceq 1530  {cab 2797  wne 3014  wral 3136  wrex 3137  wss 3934  cop 4565   cuni 4830  cres 5550   Fn wfn 6343  cfv 6348   predc-bnj14 31946   FrSe w-bnj15 31950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-uni 4831
This theorem is referenced by:  bnj1523  32331
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