Theorem frege131 41491

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 41435	. 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 4079	. . . . . . 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 3441	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
6		vex 3426	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
7	5, 6	elimasn 5986	. . . . . . . . . 10 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
8		df-br 5071	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ◡(t+‘𝑅))
9	5, 6	brcnv 5780	. . . . . . . . . 10 ⊢ (𝑀◡(t+‘𝑅)𝑏 ↔ 𝑏(t+‘𝑅)𝑀)
10	7, 8, 9	3bitr2i 298	. . . . . . . . 9 ⊢ (𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑏(t+‘𝑅)𝑀)
11	10	notbii 319	. . . . . . . 8 ⊢ (¬ 𝑏 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑏(t+‘𝑅)𝑀)
12	5, 6	elimasn 5986	. . . . . . . . 9 ⊢ (𝑏 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
13		df-br 5071	. . . . . . . . 9 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑏 ↔ ⟨𝑀, 𝑏⟩ ∈ ((t+‘𝑅) ∪ I ))
14	12, 13	bitr4i 277	. . . . . . . 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 296	. . . . . 6 ⊢ (𝑏 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏))
17		elun 4079	. . . . . . . . 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 3426	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
20	5, 19	elimasn 5986	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
21		df-br 5071	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ◡(t+‘𝑅))
22	5, 19	brcnv 5780	. . . . . . . . . . . 12 ⊢ (𝑀◡(t+‘𝑅)𝑎 ↔ 𝑎(t+‘𝑅)𝑀)
23	20, 21, 22	3bitr2i 298	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ 𝑎(t+‘𝑅)𝑀)
24	23	notbii 319	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ (◡(t+‘𝑅) “ {𝑀}) ↔ ¬ 𝑎(t+‘𝑅)𝑀)
25	5, 19	elimasn 5986	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (((t+‘𝑅) ∪ I ) “ {𝑀}) ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
26		df-br 5071	. . . . . . . . . . 11 ⊢ (𝑀((t+‘𝑅) ∪ I )𝑎 ↔ ⟨𝑀, 𝑎⟩ ∈ ((t+‘𝑅) ∪ I ))
27	25, 26	bitr4i 277	. . . . . . . . . 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 296	. . . . . . . 8 ⊢ (𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) ↔ (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))
30	29	imbi2i 335	. . . . . . 7 ⊢ ((𝑏𝑅𝑎 → 𝑎 ∈ ((◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) ↔ (𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎)))
31	30	albii 1823	. . . . . 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 1823	. . . 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 41490	. . 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 216	. 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 1537   ∈ wcel 2108   ∪ cun 3881  {csn 4558  ⟨cop 4564   class class class wbr 5070   I cid 5479  ^◡ccnv 5579   “ cima 5583  Fun wfun 6412  ‘cfv 6418  t+ctcl 14624   hereditary whe 41269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege28 41327  ax-frege31 41331  ax-frege41 41342  ax-frege52a 41354  ax-frege52c 41385  ax-frege58b 41398

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650  df-trcl 14626  df-relexp 14659  df-he 41270

This theorem is referenced by:  frege132  41492