Theorem frege131 43956

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 43900	. 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 4176	. . . . . . 7 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
3		df-or 847	. . . . . . 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 3511	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
6		vex 3492	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
7	5, 6	elimasn 6119	. . . . . . . . . 10 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
8		df-br 5167	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
9	5, 6	brcnv 5907	. . . . . . . . . 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 6119	. . . . . . . . 9 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13		df-br 5167	. . . . . . . . 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 4176	. . . . . . . . 9 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
18		df-or 847	. . . . . . . . 9 ⊢ ((𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ∨ 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) → 𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀})))
19		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
20	5, 19	elimasn 6119	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
21		df-br 5167	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
22	5, 19	brcnv 5907	. . . . . . . . . . . 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 6119	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26		df-br 5167	. . . . . . . . . . 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 1817	. . . . . 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 1817	. . . 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 43955	. . 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 846  ∀wal 1535   ∈ wcel 2108   ∪ cun 3974  {csn 4648  ⟨cop 4654   class class class wbr 5166   I cid 5592  ^◡ccnv 5699   “ cima 5703  Fun wfun 6567  ‘cfv 6573  t+ctcl 15034   hereditary whe 43734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771  ax-frege28 43792  ax-frege31 43796  ax-frege41 43807  ax-frege52a 43819  ax-frege52c 43850  ax-frege58b 43863

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-trcl 15036  df-relexp 15069  df-he 43735

This theorem is referenced by:  frege132  43957