Theorem frege131 40360

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	⊢ 𝑀 ∈ 𝑈
frege130.r	⊢ 𝑅 ∈ 𝑉

Assertion

Ref	Expression
frege131	⊢ (Fun ◡◡𝑅 → 𝑅 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.

Step	Hyp	Ref	Expression
1		frege75 40304	. 2 ⊢ (∀𝑏(𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))
2		elun 4125	. . . . . . 7 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3		df-or 844	. . . . . . 7 ⊢ ((𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
4		frege130.m	. . . . . . . . . . . 12 ⊢ 𝑀 ∈ 𝑈
5	4	elexi 3513	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
6		vex 3497	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
7	5, 6	elimasn 5954	. . . . . . . . . 10 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
8		df-br 5067	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
9	5, 6	brcnv 5753	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ 𝑏(t+‘𝑅)𝑀)
10	7, 8, 9	3bitr2i 301	. . . . . . . . 9 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
11	10	notbii 322	. . . . . . . 8 ⊢ (¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
12	5, 6	elimasn 5954	. . . . . . . . 9 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13		df-br 5067	. . . . . . . . 9 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
14	12, 13	bitr4i 280	. . . . . . . 8 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑏)
15	11, 14	imbi12i 353	. . . . . . 7 ⊢ ((¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏))
16	2, 3, 15	3bitri 299	. . . . . 6 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏))
17		elun 4125	. . . . . . . . 9 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18		df-or 844	. . . . . . . . 9 ⊢ ((𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
20	5, 19	elimasn 5954	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
21		df-br 5067	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
22	5, 19	brcnv 5753	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ 𝑎(t+‘𝑅)𝑀)
23	20, 21, 22	3bitr2i 301	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
24	23	notbii 322	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
25	5, 19	elimasn 5954	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26		df-br 5067	. . . . . . . . . . 11 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
27	25, 26	bitr4i 280	. . . . . . . . . 10 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑎)
28	24, 27	imbi12i 353	. . . . . . . . 9 ⊢ ((¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))
29	17, 18, 28	3bitri 299	. . . . . . . 8 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))
30	29	imbi2i 338	. . . . . . 7 ⊢ ((𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))
31	30	albii 1820	. . . . . 6 ⊢ (∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))
32	16, 31	imbi12i 353	. . . . 5 ⊢ ((𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) ↔ ((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))))
33	32	albii 1820	. . . 4 ⊢ (∀𝑏(𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) ↔ ∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))))
34	33	imbi1i 352	. . 3 ⊢ ((∀𝑏(𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
35		frege130.r	. . . 4 ⊢ 𝑅 ∈ 𝑉
36	4, 35	frege130 40359	. . 3 ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
37	34, 36	sylbi 219	. 2 ⊢ ((∀𝑏(𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → ∀𝑎(𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))) → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))
38	1, 37	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → 𝑅 hereditary ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843  ∀wal 1535   ∈ wcel 2114   ∪ cun 3934  {csn 4567  ⟨cop 4573   class class class wbr 5066   I cid 5459  ^◡ccnv 5554   “ cima 5558  Fun wfun 6349  ‘cfv 6355  t+ctcl 14345   hereditary whe 40138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-frege1 40156  ax-frege2 40157  ax-frege8 40175  ax-frege28 40196  ax-frege31 40200  ax-frege41 40211  ax-frege52a 40223  ax-frege52c 40254  ax-frege58b 40267

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-seq 13371  df-trcl 14347  df-relexp 14380  df-he 40139

This theorem is referenced by:  frege132  40361