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Theorem bnj865 31300
Description: Technical lemma for bnj69 31385. 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 31298 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
5 omex 8713 . . . . . . . . 9 ω ∈ V
6 difexg 4960 . . . . . . . . 9 (ω ∈ V → (ω ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . . . 8 (ω ∖ {∅}) ∈ V
83, 7eqeltri 2835 . . . . . . 7 𝐷 ∈ V
9 raleq 3277 . . . . . . . 8 (𝑧 = 𝐷 → (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
10 raleq 3277 . . . . . . . . 9 (𝑧 = 𝐷 → (∀𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1110exbidv 1999 . . . . . . . 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 7299 . . . . . . 7 (∀𝑛𝑧 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝑧𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
148, 12, 13vtocl 3399 . . . . . 6 (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
154, 14syl 17 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
16 19.37v 2075 . . . . 5 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑤𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
1715, 16mpbir 221 . . . 4 𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
18 df-ral 3055 . . . . . . . 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 2017 . . . . . . 7 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛(𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2119, 20bitr4i 267 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))))
2221exbii 1923 . . . . 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 1074 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
2524bicomi 214 . . . . . . . . 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 266 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2827albii 1896 . . . . . 6 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
2928exbii 1923 . . . . 5 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴) → (𝑛𝐷 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3022, 29bitri 264 . . . 4 (∃𝑤((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3117, 30mpbi 220 . . 3 𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
32 bnj865.5 . . . . . . 7 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
3332bicomi 214 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ 𝜒)
3433imbi1i 338 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3534albii 1896 . . . 4 (∀𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3635exbii 1923 . . 3 (∃𝑤𝑛((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
3731, 36mpbi 220 . 2 𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
38 bnj865.6 . . . . . 6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
3938rexbii 3179 . . . . 5 (∃𝑓𝑤 𝜃 ↔ ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓))
4039imbi2i 325 . . . 4 ((𝜒 → ∃𝑓𝑤 𝜃) ↔ (𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4140albii 1896 . . 3 (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4241exbii 1923 . 2 (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) ↔ ∃𝑤𝑛(𝜒 → ∃𝑓𝑤 (𝑓 Fn 𝑛𝜑𝜓)))
4337, 42mpbir 221 1 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  c0 4058  {csn 4321   ciun 4672  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7230   predc-bnj14 31063   FrSe w-bnj15 31067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-bnj17 31062  df-bnj14 31064  df-bnj13 31066  df-bnj15 31068
This theorem is referenced by:  bnj849  31302
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