Theorem frege77 40641

Description: If 𝑌 follows 𝑋 in the 𝑅-sequence, if property 𝐴 is hereditary in the 𝑅-sequence, and if every result of an application of the procedure 𝑅 to 𝑋 has the property 𝐴, then 𝑌 has property 𝐴. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege77.x	⊢ 𝑋 ∈ 𝑈
frege77.y	⊢ 𝑌 ∈ 𝑉
frege77.r	⊢ 𝑅 ∈ 𝑊
frege77.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege77	⊢ (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))

Distinct variable groups:   𝐴,𝑎   𝑅,𝑎   𝑋,𝑎

Allowed substitution hints:   𝐵(𝑎)   𝑈(𝑎)   𝑉(𝑎)   𝑊(𝑎)   𝑌(𝑎)

Proof of Theorem frege77

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege77.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2		frege77.y	. . 3 ⊢ 𝑌 ∈ 𝑉
3		frege77.r	. . 3 ⊢ 𝑅 ∈ 𝑊
4	1, 2, 3	dffrege76 40640	. 2 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌)
5		frege77.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
6	5	frege68c 40632	. . 3 ⊢ ((∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌) → (𝑋(t+‘𝑅)𝑌 → [𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓))))
7		sbcimg 3767	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 → [𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓))))
8	5, 7	ax-mp 5	. . . 4 ⊢ ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 → [𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)))
9		sbcheg 40480	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑅 hereditary 𝑓 ↔ ⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓))
10	5, 9	ax-mp 5	. . . . . 6 ⊢ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 ↔ ⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓)
11		csbconstg 3847	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑅 = 𝑅)
12	5, 11	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑅 = 𝑅
13		csbvarg 4339	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑓 = 𝐴)
14	5, 13	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴
15		heeq12 40477	. . . . . . 7 ⊢ ((⦋𝐴 / 𝑓⦌𝑅 = 𝑅 ∧ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴) → (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴))
16	12, 14, 15	mp2an 691	. . . . . 6 ⊢ (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴)
17	10, 16	bitri 278	. . . . 5 ⊢ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 ↔ 𝑅 hereditary 𝐴)
18		sbcimg 3767	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓) ↔ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → [𝐴 / 𝑓]𝑌 ∈ 𝑓)))
19	5, 18	ax-mp 5	. . . . . 6 ⊢ ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓) ↔ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → [𝐴 / 𝑓]𝑌 ∈ 𝑓))
20		sbcal 3780	. . . . . . . 8 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓))
21		sbcimg 3767	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ([𝐴 / 𝑓]𝑋𝑅𝑎 → [𝐴 / 𝑓]𝑎 ∈ 𝑓)))
22	5, 21	ax-mp 5	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ([𝐴 / 𝑓]𝑋𝑅𝑎 → [𝐴 / 𝑓]𝑎 ∈ 𝑓))
23		sbcg 3793	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎))
24	5, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎)
25		sbcel2gv 3787	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑎 ∈ 𝑓 ↔ 𝑎 ∈ 𝐴))
26	5, 25	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑓]𝑎 ∈ 𝑓 ↔ 𝑎 ∈ 𝐴)
27	24, 26	imbi12i 354	. . . . . . . . . 10 ⊢ (([𝐴 / 𝑓]𝑋𝑅𝑎 → [𝐴 / 𝑓]𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
28	22, 27	bitri 278	. . . . . . . . 9 ⊢ ([𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
29	28	albii 1821	. . . . . . . 8 ⊢ (∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
30	20, 29	bitri 278	. . . . . . 7 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
31		sbcel2gv 3787	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑌 ∈ 𝑓 ↔ 𝑌 ∈ 𝐴))
32	5, 31	ax-mp 5	. . . . . . 7 ⊢ ([𝐴 / 𝑓]𝑌 ∈ 𝑓 ↔ 𝑌 ∈ 𝐴)
33	30, 32	imbi12i 354	. . . . . 6 ⊢ (([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → [𝐴 / 𝑓]𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
34	19, 33	bitri 278	. . . . 5 ⊢ ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
35	17, 34	imbi12i 354	. . . 4 ⊢ (([𝐴 / 𝑓]𝑅 hereditary 𝑓 → [𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))
36	8, 35	bitri 278	. . 3 ⊢ ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))
37	6, 36	syl6ib 254	. 2 ⊢ ((∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌) → (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))))
38	4, 37	ax-mp 5	1 ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  [wsbc 3720  ⦋csb 3828   class class class wbr 5030  ‘cfv 6324  t+ctcl 14336   hereditary whe 40473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-frege1 40491  ax-frege2 40492  ax-frege8 40510  ax-frege52a 40558  ax-frege58b 40602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-trcl 14338  df-relexp 14371  df-he 40474

This theorem is referenced by:  frege78  40642  frege85  40649