Theorem frege131 44018

Description: If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence beginning 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 43962	. 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 4128	. . . . . . 7 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3		df-or 848	. . . . . . 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 3482	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
6		vex 3463	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
7	5, 6	elimasn 6077	. . . . . . . . . 10 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
8		df-br 5120	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
9	5, 6	brcnv 5862	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ 𝑏(t+‘𝑅)𝑀)
10	7, 8, 9	3bitr2i 299	. . . . . . . . 9 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
11	10	notbii 320	. . . . . . . 8 ⊢ (¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
12	5, 6	elimasn 6077	. . . . . . . . 9 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13		df-br 5120	. . . . . . . . 9 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
14	12, 13	bitr4i 278	. . . . . . . 8 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑏)
15	11, 14	imbi12i 350	. . . . . . 7 ⊢ ((¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏))
16	2, 3, 15	3bitri 297	. . . . . 6 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏))
17		elun 4128	. . . . . . . . 9 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18		df-or 848	. . . . . . . . 9 ⊢ ((𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
20	5, 19	elimasn 6077	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
21		df-br 5120	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
22	5, 19	brcnv 5862	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ 𝑎(t+‘𝑅)𝑀)
23	20, 21, 22	3bitr2i 299	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
24	23	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
25	5, 19	elimasn 6077	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26		df-br 5120	. . . . . . . . . . 11 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
27	25, 26	bitr4i 278	. . . . . . . . . 10 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ 𝑀((t+‘𝑅) ∪ I )𝑎)
28	24, 27	imbi12i 350	. . . . . . . . 9 ⊢ ((¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))
29	17, 18, 28	3bitri 297	. . . . . . . 8 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))
30	29	imbi2i 336	. . . . . . 7 ⊢ ((𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))
31	30	albii 1819	. . . . . 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 350	. . . . 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 1819	. . . 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 349	. . 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 44017	. . 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 217	. 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 847  ∀wal 1538   ∈ wcel 2108   ∪ cun 3924  {csn 4601  ⟨cop 4607   class class class wbr 5119   I cid 5547  ^◡ccnv 5653   “ cima 5657  Fun wfun 6525  ‘cfv 6531  t+ctcl 15004   hereditary whe 43796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-frege1 43814  ax-frege2 43815  ax-frege8 43833  ax-frege28 43854  ax-frege31 43858  ax-frege41 43869  ax-frege52a 43881  ax-frege52c 43912  ax-frege58b 43925

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-n0 12502  df-z 12589  df-uz 12853  df-seq 14020  df-trcl 15006  df-relexp 15039  df-he 43797

This theorem is referenced by:  frege132  44019