Theorem frege77 37254

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 37253	. 2 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓)) ↔ 𝑋(t+‘𝑅)𝑌)
5		frege77.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
6	5	frege68c 37245	. . 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 3444	. . . . 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 37093	. . . . . . 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 3512	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑅 = 𝑅)
12	5, 11	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑅 = 𝑅
13		csbvarg 3955	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑓⦌𝑓 = 𝐴)
14	5, 13	ax-mp 5	. . . . . . 7 ⊢ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴
15		heeq12 37090	. . . . . . 7 ⊢ ((⦋𝐴 / 𝑓⦌𝑅 = 𝑅 ∧ ⦋𝐴 / 𝑓⦌𝑓 = 𝐴) → (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴))
16	12, 14, 15	mp2an 704	. . . . . 6 ⊢ (⦋𝐴 / 𝑓⦌𝑅 hereditary ⦋𝐴 / 𝑓⦌𝑓 ↔ 𝑅 hereditary 𝐴)
17	10, 16	bitri 263	. . . . 5 ⊢ ([𝐴 / 𝑓]𝑅 hereditary 𝑓 ↔ 𝑅 hereditary 𝐴)
18		sbcimg 3444	. . . . . . 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 3452	. . . . . . . 8 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓))
21		sbcimg 3444	. . . . . . . . . . 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 3470	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎))
24	5, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑓]𝑋𝑅𝑎 ↔ 𝑋𝑅𝑎)
25		sbcel2gv 3463	. . . . . . . . . . . 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 339	. . . . . . . . . 10 ⊢ (([𝐴 / 𝑓]𝑋𝑅𝑎 → [𝐴 / 𝑓]𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
28	22, 27	bitri 263	. . . . . . . . 9 ⊢ ([𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ (𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
29	28	albii 1737	. . . . . . . 8 ⊢ (∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
30	20, 29	bitri 263	. . . . . . 7 ⊢ ([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))
31		sbcel2gv 3463	. . . . . . . 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 339	. . . . . 6 ⊢ (([𝐴 / 𝑓]∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → [𝐴 / 𝑓]𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
34	19, 33	bitri 263	. . . . 5 ⊢ ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝑌 ∈ 𝑓) ↔ (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴))
35	17, 34	imbi12i 339	. . . 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 263	. . 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 240	. 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 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   class class class wbr 4583  ‘cfv 5804  t+ctcl 13572   hereditary whe 37086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52a 37171  ax-frege58b 37215

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-trcl 13574  df-relexp 13609  df-he 37087

This theorem is referenced by:  frege78  37255  frege85  37262