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Theorem bnj865 31842
Description: Technical lemma for bnj69 31927. 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
bnj865.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj865.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj865.3 𝐷 = (ω ∖ {∅})
bnj865.5 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
bnj865.6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj865 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑤,𝐴,𝑓,𝑛   𝐷,𝑓,𝑖,𝑛   𝑤,𝐷   𝑅,𝑓,𝑖,𝑛,𝑦   𝑤,𝑅   𝑓,𝑋,𝑛,𝑤   𝜑,𝑤   𝜓,𝑤
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑤,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑤,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj865
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj865.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj865.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj865.3 . . . . . . 7 𝐷 = (ω ∖ {∅})
41, 2, 3bnj852 31840 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 omex 8894 . . . . . . . . 9 ω ∈ V
6 difexg 5081 . . . . . . . . 9 (ω ∈ V → (ω ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . . . 8 (ω ∖ {∅}) ∈ V
83, 7eqeltri 2856 . . . . . . 7 𝐷 ∈ V
9 raleq 3339 . . . . . . . 8 (𝑧 = 𝐷 → (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
10 raleq 3339 . . . . . . . . 9 (𝑧 = 𝐷 → (∀𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1110exbidv 1880 . . . . . . . 8 (𝑧 = 𝐷 → (∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
129, 11imbi12d 337 . . . . . . 7 (𝑧 = 𝐷 → ((∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
13 zfrep6 7462 . . . . . . 7 (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
148, 12, 13vtocl 3472 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
154, 14syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
16 19.37v 1948 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1715, 16mpbir 223 . . . 4 𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
18 df-ral 3087 . . . . . . . 8 (∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1918imbi2i 328 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
20 19.21v 1898 . . . . . . 7 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2119, 20bitr4i 270 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2221exbii 1810 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
23 impexp 443 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
24 df-3an 1070 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
2524bicomi 216 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
2625imbi1i 342 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2723, 26bitr3i 269 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2827albii 1782 . . . . . 6 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2928exbii 1810 . . . . 5 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3022, 29bitri 267 . . . 4 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3117, 30mpbi 222 . . 3 𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
32 bnj865.5 . . . . . . 7 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
3332bicomi 216 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ 𝜒)
3433imbi1i 342 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3534albii 1782 . . . 4 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3635exbii 1810 . . 3 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3731, 36mpbi 222 . 2 𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
38 bnj865.6 . . . . . 6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
3938rexbii 3188 . . . . 5 (∃𝑓𝑤 𝜃 ↔ ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
4039imbi2i 328 . . . 4 ((𝜒 → ∃𝑓𝑤 𝜃) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4140albii 1782 . . 3 (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4241exbii 1810 . 2 (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4337, 42mpbir 223 1 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068  wal 1505   = wceq 1507  wex 1742  wcel 2050  ∃!weu 2583  wral 3082  wrex 3083  Vcvv 3409  cdif 3820  c0 4172  {csn 4435   ciun 4786  suc csuc 6025   Fn wfn 6177  cfv 6182  ωcom 7390   predc-bnj14 31606   FrSe w-bnj15 31610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-reg 8845  ax-inf2 8892
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-om 7391  df-1o 7899  df-bnj17 31605  df-bnj14 31607  df-bnj13 31609  df-bnj15 31611
This theorem is referenced by:  bnj849  31844
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