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Theorem frege131 40218
Description: If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence begining with 𝑀 or preceeding 𝑀 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 131 of [Frege1879] p. 85. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege130.m 𝑀𝑈
frege130.r 𝑅𝑉
Assertion
Ref Expression
frege131 (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))

Proof of Theorem frege131
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frege75 40162 . 2 (∀𝑏(𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))
2 elun 4122 . . . . . . 7 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3 df-or 842 . . . . . . 7 ((𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
4 frege130.m . . . . . . . . . . . 12 𝑀𝑈
54elexi 3511 . . . . . . . . . . 11 𝑀 ∈ V
6 vex 3495 . . . . . . . . . . 11 𝑏 ∈ V
75, 6elimasn 5947 . . . . . . . . . 10 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
8 df-br 5058 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
95, 6brcnv 5746 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑀)
107, 8, 93bitr2i 300 . . . . . . . . 9 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
1110notbii 321 . . . . . . . 8 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
125, 6elimasn 5947 . . . . . . . . 9 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13 df-br 5058 . . . . . . . . 9 (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
1412, 13bitr4i 279 . . . . . . . 8 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑏)
1511, 14imbi12i 352 . . . . . . 7 ((¬ 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
162, 3, 153bitri 298 . . . . . 6 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
17 elun 4122 . . . . . . . . 9 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18 df-or 842 . . . . . . . . 9 ((𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19 vex 3495 . . . . . . . . . . . . 13 𝑎 ∈ V
205, 19elimasn 5947 . . . . . . . . . . . 12 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
21 df-br 5058 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
225, 19brcnv 5746 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎𝑎(t+‘𝑅)𝑀)
2320, 21, 223bitr2i 300 . . . . . . . . . . 11 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
2423notbii 321 . . . . . . . . . 10 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
255, 19elimasn 5947 . . . . . . . . . . 11 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26 df-br 5058 . . . . . . . . . . 11 (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
2725, 26bitr4i 279 . . . . . . . . . 10 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑎)
2824, 27imbi12i 352 . . . . . . . . 9 ((¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
2917, 18, 283bitri 298 . . . . . . . 8 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
3029imbi2i 337 . . . . . . 7 ((𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3130albii 1811 . . . . . 6 (∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3216, 31imbi12i 352 . . . . 5 ((𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) ↔ ((¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))))
3332albii 1811 . . . 4 (∀𝑏(𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) ↔ ∀𝑏((¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))))
3433imbi1i 351 . . 3 ((∀𝑏(𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (∀𝑏((¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
35 frege130.r . . . 4 𝑅𝑉
364, 35frege130 40217 . . 3 ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
3734, 36sylbi 218 . 2 ((∀𝑏(𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
381, 37ax-mp 5 1 (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 841  wal 1526  wcel 2105  cun 3931  {csn 4557  cop 4563   class class class wbr 5057   I cid 5452  ccnv 5547  cima 5551  Fun wfun 6342  cfv 6348  t+ctcl 14333   hereditary whe 39996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-frege1 40014  ax-frege2 40015  ax-frege8 40033  ax-frege28 40054  ax-frege31 40058  ax-frege41 40069  ax-frege52a 40081  ax-frege52c 40112  ax-frege58b 40125
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13358  df-trcl 14335  df-relexp 14368  df-he 39997
This theorem is referenced by:  frege132  40219
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