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Theorem frege131 38814
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 38758 . 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 3904 . . . . . . 7 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3 df-or 837 . . . . . . 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 3365 . . . . . . . . . . 11 𝑀 ∈ V
6 vex 3354 . . . . . . . . . . 11 𝑏 ∈ V
75, 6elimasn 5631 . . . . . . . . . 10 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
8 df-br 4787 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
95, 6brcnv 5443 . . . . . . . . . 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 5631 . . . . . . . . 9 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13 df-br 4787 . . . . . . . . 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 3904 . . . . . . . . 9 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18 df-or 837 . . . . . . . . 9 ((𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19 vex 3354 . . . . . . . . . . . . 13 𝑎 ∈ V
205, 19elimasn 5631 . . . . . . . . . . . 12 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
21 df-br 4787 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
225, 19brcnv 5443 . . . . . . . . . . . 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 5631 . . . . . . . . . . 11 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26 df-br 4787 . . . . . . . . . . 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 1895 . . . . . 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 1895 . . . 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 38813 . . 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 836  wal 1629  wcel 2145  cun 3721  {csn 4316  cop 4322   class class class wbr 4786   I cid 5156  ccnv 5248  cima 5252  Fun wfun 6025  cfv 6031  t+ctcl 13934   hereditary whe 38592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-frege1 38610  ax-frege2 38611  ax-frege8 38629  ax-frege28 38650  ax-frege31 38654  ax-frege41 38665  ax-frege52a 38677  ax-frege52c 38708  ax-frege58b 38721
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-z 11580  df-uz 11889  df-seq 13009  df-trcl 13936  df-relexp 13969  df-he 38593
This theorem is referenced by:  frege132  38815
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