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Theorem frege131 38605
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 38549 . 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 3786 . . . . . . 7 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3 df-or 384 . . . . . . 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 3244 . . . . . . . . . . 11 𝑀 ∈ V
6 vex 3234 . . . . . . . . . . 11 𝑏 ∈ V
75, 6elimasn 5525 . . . . . . . . . 10 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
8 df-br 4686 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
95, 6brcnv 5337 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑀)
107, 8, 93bitr2i 288 . . . . . . . . 9 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
1110notbii 309 . . . . . . . 8 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
125, 6elimasn 5525 . . . . . . . . 9 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13 df-br 4686 . . . . . . . . 9 (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
1412, 13bitr4i 267 . . . . . . . 8 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑏)
1511, 14imbi12i 339 . . . . . . 7 ((¬ 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
162, 3, 153bitri 286 . . . . . 6 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
17 elun 3786 . . . . . . . . 9 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18 df-or 384 . . . . . . . . 9 ((𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19 vex 3234 . . . . . . . . . . . . 13 𝑎 ∈ V
205, 19elimasn 5525 . . . . . . . . . . . 12 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
21 df-br 4686 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
225, 19brcnv 5337 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎𝑎(t+‘𝑅)𝑀)
2320, 21, 223bitr2i 288 . . . . . . . . . . 11 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
2423notbii 309 . . . . . . . . . 10 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
255, 19elimasn 5525 . . . . . . . . . . 11 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26 df-br 4686 . . . . . . . . . . 11 (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
2725, 26bitr4i 267 . . . . . . . . . 10 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑎)
2824, 27imbi12i 339 . . . . . . . . 9 ((¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
2917, 18, 283bitri 286 . . . . . . . 8 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
3029imbi2i 325 . . . . . . 7 ((𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3130albii 1787 . . . . . 6 (∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3216, 31imbi12i 339 . . . . 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 1787 . . . 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 338 . . 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 38604 . . 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 207 . 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 382  wal 1521  wcel 2030  cun 3605  {csn 4210  cop 4216   class class class wbr 4685   I cid 5052  ccnv 5142  cima 5146  Fun wfun 5920  cfv 5926  t+ctcl 13770   hereditary whe 38383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege28 38441  ax-frege31 38445  ax-frege41 38456  ax-frege52a 38468  ax-frege52c 38499  ax-frege58b 38512
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-trcl 13772  df-relexp 13805  df-he 38384
This theorem is referenced by:  frege132  38606
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