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Theorem bnj1307 31409
Description: Technical lemma for bnj60 31448. 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 2005 . . . . . 6 𝑥 𝑓 Fn 𝑑
5 nfra1 3129 . . . . . 6 𝑥𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)
64, 5nfan 1990 . . . . 5 𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
73, 6nfrex 3194 . . . 4 𝑥𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
87nfab 2953 . . 3 𝑥{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
91, 8nfcxfr 2946 . 2 𝑥𝐶
109nfcrii 2941 1 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635   = wceq 1637  wcel 2156  {cab 2792  wral 3096  wrex 3097   Fn wfn 6092  cfv 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102
This theorem is referenced by:  bnj1311  31410  bnj1373  31416  bnj1498  31447  bnj1525  31455  bnj1523  31457
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