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Theorem dffrege76 38053
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 37838 . . 3 ((𝐵𝑈𝐸𝑉𝑅𝑊) → (𝐵(t+‘𝑅)𝐸𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓}))
51, 2, 3, 4mp3an 1422 . 2 (𝐵(t+‘𝑅)𝐸𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓})
62elexi 3208 . . 3 𝐸 ∈ V
76elintab 4478 . 2 (𝐸 {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓))
8 imaundi 5533 . . . . . . . . 9 (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ {𝐵}) ∪ (𝑅𝑓))
98equncomi 3751 . . . . . . . 8 (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅𝑓) ∪ (𝑅 “ {𝐵}))
109sseq1i 3621 . . . . . . 7 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
11 unss 3779 . . . . . . 7 (((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ ((𝑅𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓)
1210, 11bitr4i 267 . . . . . 6 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓))
13 df-he 37887 . . . . . . . 8 (𝑅 hereditary 𝑓 ↔ (𝑅𝑓) ⊆ 𝑓)
1413bicomi 214 . . . . . . 7 ((𝑅𝑓) ⊆ 𝑓𝑅 hereditary 𝑓)
15 dfss2 3584 . . . . . . . 8 ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓))
161elexi 3208 . . . . . . . . . . . 12 𝐵 ∈ V
17 vex 3198 . . . . . . . . . . . 12 𝑎 ∈ V
1816, 17elimasn 5478 . . . . . . . . . . 11 (𝑎 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
19 df-br 4645 . . . . . . . . . . 11 (𝐵𝑅𝑎 ↔ ⟨𝐵, 𝑎⟩ ∈ 𝑅)
2018, 19bitr4i 267 . . . . . . . . . 10 (𝑎 ∈ (𝑅 “ {𝐵}) ↔ 𝐵𝑅𝑎)
2120imbi1i 339 . . . . . . . . 9 ((𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓) ↔ (𝐵𝑅𝑎𝑎𝑓))
2221albii 1745 . . . . . . . 8 (∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎𝑓) ↔ ∀𝑎(𝐵𝑅𝑎𝑎𝑓))
2315, 22bitri 264 . . . . . . 7 ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝐵𝑅𝑎𝑎𝑓))
2414, 23anbi12i 732 . . . . . 6 (((𝑅𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)))
2512, 24bitri 264 . . . . 5 ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)))
2625imbi1i 339 . . . 4 (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ ((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)) → 𝐸𝑓))
27 impexp 462 . . . 4 (((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎𝑎𝑓)) → 𝐸𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
2826, 27bitri 264 . . 3 (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
2928albii 1745 . 2 (∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓𝐸𝑓) ↔ ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)))
305, 7, 293bitrri 287 1 (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wcel 1988  {cab 2606  cun 3565  wss 3567  {csn 4168  cop 4174   cint 4466   class class class wbr 4644  cima 5107  cfv 5876  t+ctcl 13705   hereditary whe 37886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-seq 12785  df-trcl 13707  df-relexp 13742  df-he 37887
This theorem is referenced by:  frege77  38054  frege89  38066
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