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Theorem bnj579 34545
Description: Technical lemma for bnj852 34552. 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 34544 1 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1534  wcel 2099  ∃*wmo 2528  wral 3058  [wsbc 3776  cdif 3944  c0 4323  {csn 4629   ciun 4996   class class class wbr 5148   E cep 5581  suc csuc 6371   Fn wfn 6543  cfv 6548  ωcom 7870   predc-bnj14 34319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-om 7871  df-bnj17 34318
This theorem is referenced by:  bnj600  34550
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