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Theorem bnj1497 34071
Description: Technical lemma for bnj60 34073. 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
bnj1497.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1497.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1497.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
Assertion
Ref Expression
bnj1497 𝑔𝐶 Fun 𝑔
Distinct variable groups:   𝐶,𝑔   𝑓,𝑑   𝑓,𝑔
Allowed substitution hints:   𝐴(𝑥,𝑓,𝑔,𝑑)   𝐵(𝑥,𝑓,𝑔,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑓,𝑔,𝑑)   𝐺(𝑥,𝑓,𝑔,𝑑)   𝑌(𝑥,𝑓,𝑔,𝑑)

Proof of Theorem bnj1497
StepHypRef Expression
1 bnj1497.3 . . . . . 6 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
21bnj1317 33832 . . . . 5 (𝑔𝐶 → ∀𝑓 𝑔𝐶)
32nf5i 2143 . . . 4 𝑓 𝑔𝐶
4 nfv 1918 . . . 4 𝑓Fun 𝑔
53, 4nfim 1900 . . 3 𝑓(𝑔𝐶 → Fun 𝑔)
6 eleq1w 2817 . . . 4 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
7 funeq 6569 . . . 4 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
86, 7imbi12d 345 . . 3 (𝑓 = 𝑔 → ((𝑓𝐶 → Fun 𝑓) ↔ (𝑔𝐶 → Fun 𝑔)))
91bnj1436 33850 . . . . . 6 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
109bnj1299 33829 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
11 fnfun 6650 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1210, 11bnj31 33730 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1312bnj1265 33823 . . 3 (𝑓𝐶 → Fun 𝑓)
145, 8, 13chvarfv 2234 . 2 (𝑔𝐶 → Fun 𝑔)
1514rgen 3064 1 𝑔𝐶 Fun 𝑔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  wss 3949  cop 4635  cres 5679  Fun wfun 6538   Fn wfn 6539  cfv 6544   predc-bnj14 33699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546  df-fn 6547
This theorem is referenced by:  bnj60  34073
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