Theorem frege77 44388

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 44387	. 2 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌)
5		frege77.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
6	5	frege68c 44379	. . 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 3778	. . . . 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 44227	. . . . . . 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 3857	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑅 = 𝑅)
12	5, 11	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑅 = 𝑅
13		csbvarg 4375	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑓 = 𝐴)
14	5, 13	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴
15		heeq12 44224	. . . . . . 7 ⊢ ((⦋𝐴 / 𝑓⦌𝑅 = 𝑅 ∧ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴) → (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴))
16	12, 14, 15	mp2an 693	. . . . . 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 3778	. . . . . . 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 3789	. . . . . . . 8 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓))
21		sbcimg 3778	. . . . . . . . . . 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 3802	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎))
24	5, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎)
25		sbcel2gv 3796	. . . . . . . . . . . 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 1821	. . . . . . . 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 3796	. . . . . . . 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 1540   = wceq 1542   ∈ wcel 2114  [wsbc 3729  ⦋csb 3838   class class class wbr 5086  ‘cfv 6493  t+ctcl 14941   hereditary whe 44220

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-frege1 44238  ax-frege2 44239  ax-frege8 44257  ax-frege52a 44305  ax-frege58b 44349

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-n0 12432  df-z 12519  df-uz 12783  df-seq 13958  df-trcl 14943  df-relexp 14976  df-he 44221

This theorem is referenced by:  frege78  44389  frege85  44396