Theorem frege131 43983

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 43927	. 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 4162	. . . . . . 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 3500	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
6		vex 3481	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
7	5, 6	elimasn 6109	. . . . . . . . . 10 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
8		df-br 5148	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
9	5, 6	brcnv 5895	. . . . . . . . . 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 6109	. . . . . . . . 9 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13		df-br 5148	. . . . . . . . 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 4162	. . . . . . . . 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 3481	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
20	5, 19	elimasn 6109	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
21		df-br 5148	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
22	5, 19	brcnv 5895	. . . . . . . . . . . 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 6109	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26		df-br 5148	. . . . . . . . . . 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 1815	. . . . . 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 1815	. . . 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 43982	. . 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 1534   ∈ wcel 2105   ∪ cun 3960  {csn 4630  ⟨cop 4636   class class class wbr 5147   I cid 5581  ^◡ccnv 5687   “ cima 5691  Fun wfun 6556  ‘cfv 6562  t+ctcl 15020   hereditary whe 43761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-frege1 43779  ax-frege2 43780  ax-frege8 43798  ax-frege28 43819  ax-frege31 43823  ax-frege41 43834  ax-frege52a 43846  ax-frege52c 43877  ax-frege58b 43890

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 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-seq 14039  df-trcl 15022  df-relexp 15055  df-he 43762

This theorem is referenced by:  frege132  43984