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Theorem bnj579 34928
Description: Technical lemma for bnj852 34935. 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
bnj579.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj579.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj579.3 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj579 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
Distinct variable groups:   𝐴,𝑓,𝑖   𝐷,𝑓   𝑅,𝑓,𝑖   𝑓,𝑛,𝑖   𝑥,𝑓   𝑦,𝑓,𝑖
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑥,𝑦,𝑛)   𝐷(𝑥,𝑦,𝑖,𝑛)   𝑅(𝑥,𝑦,𝑛)

Proof of Theorem bnj579
Dummy variables 𝑘 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj579.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj579.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 biid 261 . 2 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
4 biid 261 . 2 ([𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜑)
5 biid 261 . 2 ([𝑔 / 𝑓]𝜓[𝑔 / 𝑓]𝜓)
6 biid 261 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
7 bnj579.3 . 2 𝐷 = (ω ∖ {∅})
8 biid 261 . 2 (((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗)))
9 biid 261 . 2 (∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))) ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ [𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓)) → (𝑓𝑗) = (𝑔𝑗))))
101, 2, 3, 4, 5, 6, 7, 8, 9bnj580 34927 1 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  [wsbc 3788  cdif 3948  c0 4333  {csn 4626   ciun 4991   class class class wbr 5143   E cep 5583  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887   predc-bnj14 34702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-om 7888  df-bnj17 34701
This theorem is referenced by:  bnj600  34933
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