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Theorem bnj1307 31905
Description: Technical lemma for bnj60 31944. 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
bnj1307.1 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1307.2 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
Assertion
Ref Expression
bnj1307 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Distinct variable groups:   𝑤,𝐵   𝑤,𝑑,𝑥   𝑥,𝑓
Allowed substitution hints:   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑤,𝑓,𝑑)   𝐺(𝑥,𝑤,𝑓,𝑑)   𝑌(𝑥,𝑤,𝑓,𝑑)

Proof of Theorem bnj1307
StepHypRef Expression
1 bnj1307.1 . . 3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2 bnj1307.2 . . . . . 6 (𝑤𝐵 → ∀𝑥 𝑤𝐵)
32nfcii 2939 . . . . 5 𝑥𝐵
4 nfv 1896 . . . . . 6 𝑥 𝑓 Fn 𝑑
5 nfra1 3188 . . . . . 6 𝑥𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)
64, 5nfan 1885 . . . . 5 𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
73, 6nfrex 3273 . . . 4 𝑥𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
87nfab 2957 . . 3 𝑥{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
91, 8nfcxfr 2949 . 2 𝑥𝐶
109nfcrii 2944 1 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1523   = wceq 1525  wcel 2083  {cab 2777  wral 3107  wrex 3108   Fn wfn 6227  cfv 6232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113
This theorem is referenced by:  bnj1311  31906  bnj1373  31912  bnj1498  31943  bnj1525  31951  bnj1523  31953
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