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Theorem bnj1497 31946
Description: Technical lemma for bnj60 31948. 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 31710 . . . . 5 (𝑔𝐶 → ∀𝑓 𝑔𝐶)
32nf5i 2117 . . . 4 𝑓 𝑔𝐶
4 nfv 1892 . . . 4 𝑓Fun 𝑔
53, 4nfim 1878 . . 3 𝑓(𝑔𝐶 → Fun 𝑔)
6 eleq1w 2865 . . . 4 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
7 funeq 6245 . . . 4 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
86, 7imbi12d 346 . . 3 (𝑓 = 𝑔 → ((𝑓𝐶 → Fun 𝑓) ↔ (𝑔𝐶 → Fun 𝑔)))
91bnj1436 31728 . . . . . 6 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
109bnj1299 31707 . . . . 5 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
11 fnfun 6323 . . . . 5 (𝑓 Fn 𝑑 → Fun 𝑓)
1210, 11bnj31 31606 . . . 4 (𝑓𝐶 → ∃𝑑𝐵 Fun 𝑓)
1312bnj1265 31701 . . 3 (𝑓𝐶 → Fun 𝑓)
145, 8, 13chvar 2369 . 2 (𝑔𝐶 → Fun 𝑔)
1514rgen 3115 1 𝑔𝐶 Fun 𝑔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  {cab 2775  wral 3105  wrex 3106  wss 3859  cop 4478  cres 5445  Fun wfun 6219   Fn wfn 6220  cfv 6225   predc-bnj14 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-in 3866  df-ss 3874  df-br 4963  df-opab 5025  df-rel 5450  df-cnv 5451  df-co 5452  df-fun 6227  df-fn 6228
This theorem is referenced by:  bnj60  31948
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