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Theorem bnj1497 31465
Description: Technical lemma for bnj60 31467. 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 31229 . . . . 5 (𝑔𝐶 → ∀𝑓 𝑔𝐶)
32nf5i 2179 . . . 4 𝑓 𝑔𝐶
4 nfv 1995 . . . 4 𝑓Fun 𝑔
53, 4nfim 1977 . . 3 𝑓(𝑔𝐶 → Fun 𝑔)
6 eleq1w 2833 . . . 4 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
7 funeq 6050 . . . 4 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
86, 7imbi12d 333 . . 3 (𝑓 = 𝑔 → ((𝑓𝐶 → Fun 𝑓) ↔ (𝑔𝐶 → Fun 𝑔)))
91bnj1436 31247 . . . . . 6 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
109bnj1299 31226 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
11 fnfun 6127 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1210, 11bnj31 31124 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1312bnj1265 31220 . . 3 (𝑓𝐶 → Fun 𝑓)
145, 8, 13chvar 2424 . 2 (𝑔𝐶 → Fun 𝑔)
1514rgen 3071 1 𝑔𝐶 Fun 𝑔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  wss 3723  cop 4323  cres 5252  Fun wfun 6024   Fn wfn 6025  cfv 6030   predc-bnj14 31093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-in 3730  df-ss 3737  df-br 4788  df-opab 4848  df-rel 5257  df-cnv 5258  df-co 5259  df-fun 6032  df-fn 6033
This theorem is referenced by:  bnj60  31467
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