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Theorem bnj849 31316
Description: Technical lemma for bnj69 31399. 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 31314 . . 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 31315 . . . . . . 7 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
12 df-rex 3102 . . . . . . . . 9 (∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′)))
13 19.29 1963 . . . . . . . . . . 11 ((∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ∧ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → ∃𝑛((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))))
14 an12 627 . . . . . . . . . . . . 13 (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) ↔ (𝑛𝐷 ∧ ((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′))))
15 df-3an 1102 . . . . . . . . . . . . . . . 16 ((𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
161anbi1i 612 . . . . . . . . . . . . . . . 16 ((𝜏𝑛𝐷) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑛𝐷))
1715, 5, 163bitr4i 294 . . . . . . . . . . . . . . 15 (𝜒 ↔ (𝜏𝑛𝐷))
18 id 22 . . . . . . . . . . . . . . . . 17 (𝜒𝜒)
19 bnj849.9 . . . . . . . . . . . . . . . . . . . 20 (𝜃′[𝑔 / 𝑓]𝜃)
206, 9, 10, 19bnj581 31299 . . . . . . . . . . . . . . . . . . . 20 (𝜃′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
2119, 20bitr3i 268 . . . . . . . . . . . . . . . . . . 19 ([𝑔 / 𝑓]𝜃 ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
222, 3, 4, 5, 6bnj864 31313 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → ∃!𝑓𝜃)
23 df-rex 3102 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑓𝑤 𝜃 ↔ ∃𝑓(𝑓𝑤𝜃))
24 exancom 1947 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑓(𝑓𝑤𝜃) ↔ ∃𝑓(𝜃𝑓𝑤))
2523, 24sylbb 210 . . . . . . . . . . . . . . . . . . . 20 (∃𝑓𝑤 𝜃 → ∃𝑓(𝜃𝑓𝑤))
26 nfeu1 2642 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓∃!𝑓𝜃
27 nfe1 2194 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓𝑓(𝜃𝑓𝑤)
2826, 27nfan 1990 . . . . . . . . . . . . . . . . . . . . . 22 𝑓(∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤))
29 nfsbc1v 3653 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓[𝑔 / 𝑓]𝜃
30 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 𝑔𝑤
3129, 30nfim 1987 . . . . . . . . . . . . . . . . . . . . . 22 𝑓([𝑔 / 𝑓]𝜃𝑔𝑤)
3228, 31nfim 1987 . . . . . . . . . . . . . . . . . . . . 21 𝑓((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
33 sbceq1a 3644 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑔 → (𝜃[𝑔 / 𝑓]𝜃))
34 elequ1 2163 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑔 → (𝑓𝑤𝑔𝑤))
3533, 34imbi12d 335 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑔 → ((𝜃𝑓𝑤) ↔ ([𝑔 / 𝑓]𝜃𝑔𝑤)))
3635imbi2d 331 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑔 → (((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → (𝜃𝑓𝑤)) ↔ ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))))
37 eupick 2700 . . . . . . . . . . . . . . . . . . . . 21 ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → (𝜃𝑓𝑤))
3832, 36, 37chvar 2436 . . . . . . . . . . . . . . . . . . . 20 ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃𝑓𝑤)) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
3922, 25, 38syl2an 585 . . . . . . . . . . . . . . . . . . 19 ((𝜒 ∧ ∃𝑓𝑤 𝜃) → ([𝑔 / 𝑓]𝜃𝑔𝑤))
4021, 39syl5bir 234 . . . . . . . . . . . . . . . . . 18 ((𝜒 ∧ ∃𝑓𝑤 𝜃) → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤))
4140ex 399 . . . . . . . . . . . . . . . . 17 (𝜒 → (∃𝑓𝑤 𝜃 → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤)))
4218, 41embantd 59 . . . . . . . . . . . . . . . 16 (𝜒 → ((𝜒 → ∃𝑓𝑤 𝜃) → ((𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤)))
4342impd 398 . . . . . . . . . . . . . . 15 (𝜒 → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
4417, 43sylbir 226 . . . . . . . . . . . . . 14 ((𝜏𝑛𝐷) → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
4544expimpd 443 . . . . . . . . . . . . 13 (𝜏 → ((𝑛𝐷 ∧ ((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4614, 45syl5bi 233 . . . . . . . . . . . 12 (𝜏 → (((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4746exlimdv 2024 . . . . . . . . . . 11 (𝜏 → (∃𝑛((𝜒 → ∃𝑓𝑤 𝜃) ∧ (𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4813, 47syl5 34 . . . . . . . . . 10 (𝜏 → ((∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) ∧ ∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′))) → 𝑔𝑤))
4948expdimp 442 . . . . . . . . 9 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → (∃𝑛(𝑛𝐷 ∧ (𝑔 Fn 𝑛𝜑′𝜓′)) → 𝑔𝑤))
5012, 49syl5bi 233 . . . . . . . 8 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → (∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′) → 𝑔𝑤))
5150abssdv 3873 . . . . . . 7 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)} ⊆ 𝑤)
5211, 51syl5eqss 3846 . . . . . 6 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → 𝐵𝑤)
53 vex 3394 . . . . . . 7 𝑤 ∈ V
5453ssex 4997 . . . . . 6 (𝐵𝑤𝐵 ∈ V)
5552, 54syl 17 . . . . 5 ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓𝑤 𝜃)) → 𝐵 ∈ V)
5655ex 399 . . . 4 (𝜏 → (∀𝑛(𝜒 → ∃𝑓𝑤 𝜃) → 𝐵 ∈ V))
5756exlimdv 2024 . . 3 (𝜏 → (∃𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃) → 𝐵 ∈ V))
587, 57mpi 20 . 2 (𝜏𝐵 ∈ V)
591, 58sylbir 226 1 ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  {cab 2792  wral 3096  wrex 3097  Vcvv 3391  [wsbc 3633  cdif 3766  wss 3769  c0 4116  {csn 4370   ciun 4712  suc csuc 5938   Fn wfn 6092  cfv 6097  ωcom 7291   predc-bnj14 31078   FrSe w-bnj15 31082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-reg 8732  ax-inf2 8781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-om 7292  df-1o 7792  df-bnj17 31077  df-bnj14 31079  df-bnj13 31081  df-bnj15 31083
This theorem is referenced by:  bnj893  31319
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