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Theorem bnj865 33599
Description: Technical lemma for bnj69 33686. 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 33597 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 omex 9587 . . . . . . . . 9 ω ∈ V
6 difexg 5288 . . . . . . . . 9 (ω ∈ V → (ω ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . . . 8 (ω ∖ {∅}) ∈ V
83, 7eqeltri 2830 . . . . . . 7 𝐷 ∈ V
9 raleq 3308 . . . . . . . 8 (𝑧 = 𝐷 → (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
10 raleq 3308 . . . . . . . . 9 (𝑧 = 𝐷 → (∀𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1110exbidv 1925 . . . . . . . 8 (𝑧 = 𝐷 → (∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
129, 11imbi12d 345 . . . . . . 7 (𝑧 = 𝐷 → ((∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
13 zfrep6 7891 . . . . . . 7 (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
148, 12, 13vtocl 3520 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
154, 14syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
16 19.37v 1996 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1715, 16mpbir 230 . . . 4 𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
18 df-ral 3062 . . . . . . . 8 (∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1918imbi2i 336 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
20 19.21v 1943 . . . . . . 7 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2119, 20bitr4i 278 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2221exbii 1851 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
23 impexp 452 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
24 df-3an 1090 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
2524bicomi 223 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
2625imbi1i 350 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2723, 26bitr3i 277 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2827albii 1822 . . . . . 6 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2928exbii 1851 . . . . 5 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3022, 29bitri 275 . . . 4 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3117, 30mpbi 229 . . 3 𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
32 bnj865.5 . . . . . . 7 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
3332bicomi 223 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ 𝜒)
3433imbi1i 350 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3534albii 1822 . . . 4 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3635exbii 1851 . . 3 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3731, 36mpbi 229 . 2 𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
38 bnj865.6 . . . . . 6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
3938rexbii 3094 . . . . 5 (∃𝑓𝑤 𝜃 ↔ ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
4039imbi2i 336 . . . 4 ((𝜒 → ∃𝑓𝑤 𝜃) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4140albii 1822 . . 3 (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4241exbii 1851 . 2 (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4337, 42mpbir 230 1 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2563  wral 3061  wrex 3070  Vcvv 3447  cdif 3911  c0 4286  {csn 4590   ciun 4958  suc csuc 6323   Fn wfn 6495  cfv 6500  ωcom 7806   predc-bnj14 33364   FrSe w-bnj15 33368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-bnj17 33363  df-bnj14 33365  df-bnj13 33367  df-bnj15 33369
This theorem is referenced by:  bnj849  33601
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