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Theorem bnj1497 30863
Description: Technical lemma for bnj60 30865. 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 30627 . . . . 5 (𝑔𝐶 → ∀𝑓 𝑔𝐶)
32nf5i 2021 . . . 4 𝑓 𝑔𝐶
4 nfv 1840 . . . 4 𝑓Fun 𝑔
53, 4nfim 1822 . . 3 𝑓(𝑔𝐶 → Fun 𝑔)
6 eleq1 2686 . . . 4 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
7 funeq 5872 . . . 4 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
86, 7imbi12d 334 . . 3 (𝑓 = 𝑔 → ((𝑓𝐶 → Fun 𝑓) ↔ (𝑔𝐶 → Fun 𝑔)))
91bnj1436 30645 . . . . . 6 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
109bnj1299 30624 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
11 fnfun 5951 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1210, 11bnj31 30520 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1312bnj1265 30618 . . 3 (𝑓𝐶 → Fun 𝑓)
145, 8, 13chvar 2261 . 2 (𝑔𝐶 → Fun 𝑔)
1514rgen 2917 1 𝑔𝐶 Fun 𝑔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  wss 3559  cop 4159  cres 5081  Fun wfun 5846   Fn wfn 5847  cfv 5852   predc-bnj14 30488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-in 3566  df-ss 3573  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5854  df-fn 5855
This theorem is referenced by:  bnj60  30865
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