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Theorem bnj1497 31456
Description: Technical lemma for bnj60 31458. 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 31220 . . . . 5 (𝑔𝐶 → ∀𝑓 𝑔𝐶)
32nf5i 2191 . . . 4 𝑓 𝑔𝐶
4 nfv 2005 . . . 4 𝑓Fun 𝑔
53, 4nfim 1987 . . 3 𝑓(𝑔𝐶 → Fun 𝑔)
6 eleq1w 2875 . . . 4 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
7 funeq 6124 . . . 4 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
86, 7imbi12d 335 . . 3 (𝑓 = 𝑔 → ((𝑓𝐶 → Fun 𝑓) ↔ (𝑔𝐶 → Fun 𝑔)))
91bnj1436 31238 . . . . . 6 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
109bnj1299 31217 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
11 fnfun 6202 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1210, 11bnj31 31116 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1312bnj1265 31211 . . 3 (𝑓𝐶 → Fun 𝑓)
145, 8, 13chvar 2438 . 2 (𝑔𝐶 → Fun 𝑔)
1514rgen 3117 1 𝑔𝐶 Fun 𝑔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  {cab 2799  wral 3103  wrex 3104  wss 3776  cop 4383  cres 5320  Fun wfun 6098   Fn wfn 6099  cfv 6104   predc-bnj14 31085
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 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
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 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-in 3783  df-ss 3790  df-br 4852  df-opab 4914  df-rel 5325  df-cnv 5326  df-co 5327  df-fun 6106  df-fn 6107
This theorem is referenced by:  bnj60  31458
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