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Theorem bnj1307 32441
 Description: Technical lemma for bnj60 32480. 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 2940 . . . . 5 𝑥𝐵
4 nfv 1915 . . . . . 6 𝑥 𝑓 Fn 𝑑
5 nfra1 3183 . . . . . 6 𝑥𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)
64, 5nfan 1900 . . . . 5 𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
73, 6nfrex 3268 . . . 4 𝑥𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
87nfab 2961 . . 3 𝑥{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
91, 8nfcxfr 2953 . 2 𝑥𝐶
109nfcrii 2948 1 (𝑤𝐶 → ∀𝑥 𝑤𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   Fn wfn 6322  ‘cfv 6327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112 This theorem is referenced by:  bnj1311  32442  bnj1373  32448  bnj1498  32479  bnj1525  32487  bnj1523  32489
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