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Theorem dffrege76 37146
Description: If from the two propositions that every result of an application of the procedure 𝑅 to 𝐵 has property 𝑓 and that property 𝑓 is hereditary in the 𝑅-sequence, it can be inferred, whatever 𝑓 may be, that 𝐸 has property 𝑓, then we say 𝐸 follows 𝐵 in the 𝑅-sequence. Definition 76 of [Frege1879] p. 60.

Each of 𝐵, 𝐸 and 𝑅 must be sets. (Contributed by RP, 2-Jul-2020.)

Hypotheses
Ref Expression
frege76.b 𝐵𝑈
frege76.e 𝐸𝑉
frege76.r 𝑅𝑊
Assertion
Ref Expression
dffrege76 (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
Distinct variable groups:   𝑓,𝑎,𝐵   𝑓,𝐸   𝑅,𝑎,𝑓   𝑈,𝑓   𝑓,𝑉   𝑓,𝑊
Allowed substitution hints:   𝑈(𝑎)   𝐸(𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem dffrege76
StepHypRef Expression
1 frege76.b . . 3 𝐵𝑈
2 frege76.e . . 3 𝐸𝑉
3 frege76.r . . 3 𝑅𝑊
4 brtrclfv2 36931 . . 3 ((𝐵𝑈𝐸𝑉𝑅𝑊) → (𝐵(t+‘𝑅)𝐸𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓}))
51, 2, 3, 4mp3an 1415 . 2 (𝐵(t+‘𝑅)𝐸𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓})
62elexi 3090 . . 3 𝐸 ∈ V
76elintab 4320 . 2 (𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓))
8 imaundi 5354 . . . . . . . . 9 (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ {𝐵}) ∪ (𝑅𝑓))
98equncomi 3625 . . . . . . . 8 (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅𝑓) ∪ (𝑅 “ {𝐵}))
109sseq1i 3496 . . . . . . 7 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
11 unss 3653 . . . . . . 7 (((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ ((𝑅𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
1210, 11bitr4i 265 . . . . . 6 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓))
13 df-he 36980 . . . . . . . 8 (𝑅 hereditary 𝑓 ↔ (𝑅𝑓) ⊆ 𝑓)
1413bicomi 212 . . . . . . 7 ((𝑅𝑓) ⊆ 𝑓𝑅 hereditary 𝑓)
15 dfss2 3461 . . . . . . . 8 ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓))
161elexi 3090 . . . . . . . . . . . 12 𝐵 ∈ V
17 vex 3080 . . . . . . . . . . . 12 𝑎 ∈ V
1816, 17elimasn 5300 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
19 df-br 4482 . . . . . . . . . . 11 (𝐵𝑅𝑎 ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
2018, 19bitr4i 265 . . . . . . . . . 10 (𝑎 ∈ (𝑅 “ {𝐵}) ↔ 𝐵𝑅𝑎)
2120imbi1i 337 . . . . . . . . 9 ((𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓) ↔ (𝐵𝑅𝑎𝑎𝑓))
2221albii 1722 . . . . . . . 8 (∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓) ↔ ∀𝑎(𝐵𝑅𝑎𝑎𝑓))
2315, 22bitri 262 . . . . . . 7 ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝐵𝑅𝑎𝑎𝑓))
2414, 23anbi12i 728 . . . . . 6 (((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)))
2512, 24bitri 262 . . . . 5 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)))
2625imbi1i 337 . . . 4 (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ ((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)) → 𝐸𝑓))
27 impexp 460 . . . 4 (((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)) → 𝐸𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
2826, 27bitri 262 . . 3 (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
2928albii 1722 . 2 (∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
305, 7, 293bitrri 285 1 (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1938  {cab 2500  cun 3442  wss 3444  {csn 4028  cop 4034   cint 4308   class class class wbr 4481  cima 4935  cfv 5689  t+ctcl 13427   hereditary whe 36979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-er 7503  df-en 7716  df-dom 7717  df-sdom 7718  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-2 10832  df-n0 11046  df-z 11117  df-uz 11424  df-seq 12529  df-trcl 13429  df-relexp 13464  df-he 36980
This theorem is referenced by:  frege77  37147  frege89  37159
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