Theorem frege77 43964

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 43963	. 2 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌)
5		frege77.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
6	5	frege68c 43955	. . 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 3814	. . . . 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 43803	. . . . . . 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 3893	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑅 = 𝑅)
12	5, 11	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑅 = 𝑅
13		csbvarg 4409	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑓 = 𝐴)
14	5, 13	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴
15		heeq12 43800	. . . . . . 7 ⊢ ((⦋𝐴 / 𝑓⦌𝑅 = 𝑅 ∧ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴) → (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴))
16	12, 14, 15	mp2an 692	. . . . . 6 ⊢ (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴)
17	10, 16	bitri 275	. . . . 5 ⊢ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 ↔ 𝑅 hereditary 𝐴)
18		sbcimg 3814	. . . . . . 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 3825	. . . . . . . 8 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓))
21		sbcimg 3814	. . . . . . . . . . 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 3838	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎))
24	5, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎)
25		sbcel2gv 3832	. . . . . . . . . . . 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 350	. . . . . . . . . 10 ⊢ (([𝐴 / 𝑓]𝑋𝑅𝑎 → [𝐴 / 𝑓]𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
28	22, 27	bitri 275	. . . . . . . . 9 ⊢ ([𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
29	28	albii 1819	. . . . . . . 8 ⊢ (∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
30	20, 29	bitri 275	. . . . . . 7 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
31		sbcel2gv 3832	. . . . . . . 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 350	. . . . . 6 ⊢ (([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → [𝐴 / 𝑓]𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
34	19, 33	bitri 275	. . . . 5 ⊢ ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
35	17, 34	imbi12i 350	. . . 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 275	. . 3 ⊢ ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))
37	6, 36	imbitrdi 251	. 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 206  ∀wal 1538   = wceq 1540   ∈ wcel 2108  [wsbc 3765  ⦋csb 3874   class class class wbr 5119  ‘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-frege52a 43881  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:  frege78  43965  frege85  43972