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Theorem frege131 37104
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 37048 . 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 3714 . . . . . . 7 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3 df-or 383 . . . . . . 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 3185 . . . . . . . . . . 11 𝑀 ∈ V
6 vex 3175 . . . . . . . . . . 11 𝑏 ∈ V
75, 6elimasn 5396 . . . . . . . . . 10 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
8 df-br 4578 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ (t+‘𝑅))
95, 6brcnv 5215 . . . . . . . . . 10 (𝑀(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑀)
107, 8, 93bitr2i 286 . . . . . . . . 9 (𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
1110notbii 308 . . . . . . . 8 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
125, 6elimasn 5396 . . . . . . . . 9 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13 df-br 4578 . . . . . . . . 9 (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
1412, 13bitr4i 265 . . . . . . . 8 (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑏)
1511, 14imbi12i 338 . . . . . . 7 ((¬ 𝑏 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
162, 3, 153bitri 284 . . . . . 6 (𝑏 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏))
17 elun 3714 . . . . . . . . 9 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18 df-or 383 . . . . . . . . 9 ((𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19 vex 3175 . . . . . . . . . . . . 13 𝑎 ∈ V
205, 19elimasn 5396 . . . . . . . . . . . 12 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
21 df-br 4578 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ (t+‘𝑅))
225, 19brcnv 5215 . . . . . . . . . . . 12 (𝑀(t+‘𝑅)𝑎𝑎(t+‘𝑅)𝑀)
2320, 21, 223bitr2i 286 . . . . . . . . . . 11 (𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
2423notbii 308 . . . . . . . . . 10 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
255, 19elimasn 5396 . . . . . . . . . . 11 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26 df-br 4578 . . . . . . . . . . 11 (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
2725, 26bitr4i 265 . . . . . . . . . 10 (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑎)
2824, 27imbi12i 338 . . . . . . . . 9 ((¬ 𝑎 ∈ ((t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
2917, 18, 283bitri 284 . . . . . . . 8 (𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))
3029imbi2i 324 . . . . . . 7 ((𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3130albii 1736 . . . . . 6 (∀𝑎(𝑏𝑅𝑎𝑎 ∈ (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎)))
3216, 31imbi12i 338 . . . . 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 1736 . . . 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 337 . . 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 37103 . . 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 205 . 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 381  wal 1472  wcel 1976  cun 3537  {csn 4124  cop 4130   class class class wbr 4577   I cid 4938  ccnv 5027  cima 5031  Fun wfun 5784  cfv 5790  t+ctcl 13518   hereditary whe 36882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-frege1 36900  ax-frege2 36901  ax-frege8 36919  ax-frege28 36940  ax-frege31 36944  ax-frege41 36955  ax-frege52a 36967  ax-frege52c 36998  ax-frege58b 37011
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  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-pred 5583  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-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-seq 12619  df-trcl 13520  df-relexp 13555  df-he 36883
This theorem is referenced by:  frege132  37105
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