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Theorem bnj849 30043
Description: Technical lemma for bnj69 30126. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj849.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj849.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj849.3 𝐷 = (ω ∖ {∅})
bnj849.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj849.5 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
bnj849.6 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj849.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj849.8 (𝜓′[𝑔 / 𝑓]𝜓)
bnj849.9 (𝜃′[𝑔 / 𝑓]𝜃)
bnj849.10 (𝜏 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
Assertion
Ref Expression
bnj849 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐵,𝑔   𝐷,𝑓,𝑔,𝑛   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑛   𝜒,𝑓,𝑔   𝜑,𝑔   𝜓,𝑔   𝜏,𝑔,𝑛   𝜃,𝑔
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑛)   𝜏(𝑦,𝑓,𝑖)   𝐴(𝑔)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑅(𝑔)   𝑋(𝑦,𝑔,𝑖)   𝜑′(𝑦,𝑓,𝑔,𝑖,𝑛)   𝜓′(𝑦,𝑓,𝑔,𝑖,𝑛)   𝜃′(𝑦,𝑓,𝑔,𝑖,𝑛)

Proof of Theorem bnj849
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj849.10 . 2 (𝜏 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj849.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3 bnj849.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj849.3 . . . 4 𝐷 = (ω ∖ {∅})
5 bnj849.5 . . . 4 (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))
6 bnj849.6 . . . 4 (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))
72, 3, 4, 5, 6bnj865 30041 . . 3 𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)
8 bnj849.4 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
9 bnj849.7 . . . . . . . 8 (𝜑′[𝑔 / 𝑓]𝜑)
10 bnj849.8 . . . . . . . 8 (𝜓′[𝑔 / 𝑓]𝜓)
118, 9, 10bnj873 30042 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
12 df-rex 2902 . . . . . . . . 9 (∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′)))
13 19.29 1789 . . . . . . . . . . 11 ((∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ∧ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → ∃𝑛((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))))
14 an12 834 . . . . . . . . . . . . 13 (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) ↔ (𝑛𝐷 ∧ ((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′))))
15 df-3an 1033 . . . . . . . . . . . . . . . 16 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
161anbi1i 727 . . . . . . . . . . . . . . . 16 ((𝜏𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
1715, 5, 163bitr4i 291 . . . . . . . . . . . . . . 15 (𝜒 ↔ (𝜏𝑛𝐷))
18 id 22 . . . . . . . . . . . . . . . . 17 (𝜒𝜒)
19 bnj849.9 . . . . . . . . . . . . . . . . . . . 20 (𝜃′[𝑔 / 𝑓]𝜃)
206, 9, 10, 19bnj581 30026 . . . . . . . . . . . . . . . . . . . 20 (𝜃′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
2119, 20bitr3i 265 . . . . . . . . . . . . . . . . . . 19 ([𝑔 / 𝑓]𝜃 ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
222, 3, 4, 5, 6bnj864 30040 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → ∃!𝑓𝜃)
23 df-rex 2902 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑓𝑤 𝜃 ↔ ∃𝑓(𝑓𝑤𝜃))
24 exancom 1774 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑓(𝑓𝑤𝜃) ↔ ∃𝑓(𝜃𝑓𝑤))
2523, 24sylbb 208 . . . . . . . . . . . . . . . . . . . 20 (∃𝑓𝑤 𝜃 → ∃𝑓(𝜃𝑓𝑤))
26 nfeu1 2468 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓∃!𝑓𝜃
27 nfe1 2014 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓𝑓(𝜃𝑓𝑤)
2826, 27nfan 1816 . . . . . . . . . . . . . . . . . . . . . 22 𝑓(∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤))
29 nfsbc1v 3422 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓[𝑔 / 𝑓]𝜃
30 nfv 1830 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 𝑔𝑤
3129, 30nfim 1813 . . . . . . . . . . . . . . . . . . . . . 22 𝑓([𝑔 / 𝑓]𝜃𝑔𝑤)
3228, 31nfim 1813 . . . . . . . . . . . . . . . . . . . . 21 𝑓((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
33 sbceq1a 3413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑔 → (𝜃[𝑔 / 𝑓]𝜃))
34 elequ1 1984 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑔 → (𝑓𝑤𝑔𝑤))
3533, 34imbi12d 333 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → ((𝜃𝑓𝑤) ↔ ([𝑔 / 𝑓]𝜃𝑔𝑤)))
3635imbi2d 329 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → (𝜃𝑓𝑤)) ↔ ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))))
37 eupick 2524 . . . . . . . . . . . . . . . . . . . . 21 ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → (𝜃𝑓𝑤))
3832, 36, 37chvar 2250 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
3922, 25, 38syl2an 493 . . . . . . . . . . . . . . . . . . 19 ((𝜒 ∧ ∃𝑓𝑤 𝜃) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
4021, 39syl5bir 232 . . . . . . . . . . . . . . . . . 18 ((𝜒 ∧ ∃𝑓𝑤 𝜃) → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤))
4140ex 449 . . . . . . . . . . . . . . . . 17 (𝜒 → (∃𝑓𝑤 𝜃 → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤)))
4218, 41embantd 57 . . . . . . . . . . . . . . . 16 (𝜒 → ((𝜒 → ∃𝑓𝑤 𝜃) → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤)))
4342impd 446 . . . . . . . . . . . . . . 15 (𝜒 → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
4417, 43sylbir 224 . . . . . . . . . . . . . 14 ((𝜏𝑛𝐷) → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
4544expimpd 627 . . . . . . . . . . . . 13 (𝜏 → ((𝑛𝐷 ∧ ((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4614, 45syl5bi 231 . . . . . . . . . . . 12 (𝜏 → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4746exlimdv 1848 . . . . . . . . . . 11 (𝜏 → (∃𝑛((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4813, 47syl5 33 . . . . . . . . . 10 (𝜏 → ((∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ∧ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4948expdimp 452 . . . . . . . . 9 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → (∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
5012, 49syl5bi 231 . . . . . . . 8 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → (∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤))
5150abssdv 3639 . . . . . . 7 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)} ⊆ 𝑤)
5211, 51syl5eqss 3612 . . . . . 6 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → 𝐵𝑤)
53 vex 3176 . . . . . . 7 𝑤 ∈ V
5453ssex 4725 . . . . . 6 (𝐵𝑤𝐵 ∈ V)
5552, 54syl 17 . . . . 5 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → 𝐵 ∈ V)
5655ex 449 . . . 4 (𝜏 → (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) → 𝐵 ∈ V))
5756exlimdv 1848 . . 3 (𝜏 → (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) → 𝐵 ∈ V))
587, 57mpi 20 . 2 (𝜏𝐵 ∈ V)
591, 58sylbir 224 1 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
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  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  [wsbc 3402  cdif 3537  wss 3540  c0 3874  {csn 4125   ciun 4450  suc csuc 5628   Fn wfn 5785  cfv 5790  ωcom 6935   predc-bnj14 29801   FrSe w-bnj15 29805
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-reg 8358  ax-inf2 8399
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6936  df-1o 7425  df-bnj17 29800  df-bnj14 29802  df-bnj13 29804  df-bnj15 29806
This theorem is referenced by:  bnj893  30046
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