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Theorem bnj865 30247
Description: Technical lemma for bnj69 30332. 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 30245 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 omex 8423 . . . . . . . . 9 ω ∈ V
6 difexg 4735 . . . . . . . . 9 (ω ∈ V → (ω ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . . . 8 (ω ∖ {∅}) ∈ V
83, 7eqeltri 2684 . . . . . . 7 𝐷 ∈ V
9 raleq 3115 . . . . . . . 8 (𝑧 = 𝐷 → (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
10 raleq 3115 . . . . . . . . 9 (𝑧 = 𝐷 → (∀𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1110exbidv 1837 . . . . . . . 8 (𝑧 = 𝐷 → (∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
129, 11imbi12d 333 . . . . . . 7 (𝑧 = 𝐷 → ((∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
13 zfrep6 7027 . . . . . . 7 (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
148, 12, 13vtocl 3232 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
154, 14syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
16 19.37v 1897 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1715, 16mpbir 220 . . . 4 𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
18 df-ral 2901 . . . . . . . 8 (∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1918imbi2i 325 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
20 19.21v 1855 . . . . . . 7 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2119, 20bitr4i 266 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2221exbii 1764 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
23 impexp 461 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
24 df-3an 1033 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
2524bicomi 213 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
2625imbi1i 338 . . . . . . . 8 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2723, 26bitr3i 265 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2827albii 1737 . . . . . 6 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2928exbii 1764 . . . . 5 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3022, 29bitri 263 . . . 4 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3117, 30mpbi 219 . . 3 𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
32 bnj865.5 . . . . . . 7 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
3332bicomi 213 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ 𝜒)
3433imbi1i 338 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3534albii 1737 . . . 4 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3635exbii 1764 . . 3 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3731, 36mpbi 219 . 2 𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
38 bnj865.6 . . . . . 6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
3938rexbii 3023 . . . . 5 (∃𝑓𝑤 𝜃 ↔ ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
4039imbi2i 325 . . . 4 ((𝜒 → ∃𝑓𝑤 𝜃) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4140albii 1737 . . 3 (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4241exbii 1764 . 2 (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4337, 42mpbir 220 1 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wex 1695  wcel 1977  ∃!weu 2458  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  c0 3874  {csn 4125   ciun 4455  suc csuc 5642   Fn wfn 5799  cfv 5804  ωcom 6957   predc-bnj14 30007   FrSe w-bnj15 30011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012
This theorem is referenced by:  bnj849  30249
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