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Theorem bnj1514 31463
Description: Technical lemma for bnj1500 31468. 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
bnj1514.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1514.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1514.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
Assertion
Ref Expression
bnj1514 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
Distinct variable groups:   𝑥,𝐴   𝐺,𝑑   𝑌,𝑑   𝑓,𝑑,𝑥
Allowed substitution hints:   𝐴(𝑓,𝑑)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑓)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1514
StepHypRef Expression
1 bnj1514.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
21bnj1436 31242 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
3 df-rex 3066 . . . . 5 (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑(𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
4 3anass 1079 . . . . 5 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑑𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
53, 4bnj133 31127 . . . 4 (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
62, 5sylib 208 . . 3 (𝑓𝐶 → ∃𝑑(𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
7 simp3 1131 . . . 4 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) → ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))
8 fndm 6130 . . . . . 6 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
983ad2ant2 1127 . . . . 5 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) → dom 𝑓 = 𝑑)
109raleqdv 3292 . . . 4 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) → (∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
117, 10mpbird 247 . . 3 ((𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
126, 11bnj593 31147 . 2 (𝑓𝐶 → ∃𝑑𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
1312bnj937 31174 1 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wral 3060  wrex 3061  wss 3721  cop 4320  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031   predc-bnj14 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-fn 6034
This theorem is referenced by:  bnj1501  31467
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