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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffrege76 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frege76.b | ⊢ 𝐵 ∈ 𝑈 |
frege76.e | ⊢ 𝐸 ∈ 𝑉 |
frege76.r | ⊢ 𝑅 ∈ 𝑊 |
Ref | Expression |
---|---|
dffrege76 | ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege76.b | . . 3 ⊢ 𝐵 ∈ 𝑈 | |
2 | frege76.e | . . 3 ⊢ 𝐸 ∈ 𝑉 | |
3 | frege76.r | . . 3 ⊢ 𝑅 ∈ 𝑊 | |
4 | brtrclfv2 41335 | . . 3 ⊢ ((𝐵 ∈ 𝑈 ∧ 𝐸 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐵(t+‘𝑅)𝐸 ↔ 𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓})) | |
5 | 1, 2, 3, 4 | mp3an 1460 | . 2 ⊢ (𝐵(t+‘𝑅)𝐸 ↔ 𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓}) |
6 | 2 | elexi 3451 | . . 3 ⊢ 𝐸 ∈ V |
7 | 6 | elintab 4890 | . 2 ⊢ (𝐸 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓)) |
8 | imaundi 6053 | . . . . . . . . 9 ⊢ (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ {𝐵}) ∪ (𝑅 “ 𝑓)) | |
9 | 8 | equncomi 4089 | . . . . . . . 8 ⊢ (𝑅 “ ({𝐵} ∪ 𝑓)) = ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵})) |
10 | 9 | sseq1i 3949 | . . . . . . 7 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓) |
11 | unss 4118 | . . . . . . 7 ⊢ (((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ ((𝑅 “ 𝑓) ∪ (𝑅 “ {𝐵})) ⊆ 𝑓) | |
12 | 10, 11 | bitr4i 277 | . . . . . 6 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓)) |
13 | df-he 41381 | . . . . . . . 8 ⊢ (𝑅 hereditary 𝑓 ↔ (𝑅 “ 𝑓) ⊆ 𝑓) | |
14 | 13 | bicomi 223 | . . . . . . 7 ⊢ ((𝑅 “ 𝑓) ⊆ 𝑓 ↔ 𝑅 hereditary 𝑓) |
15 | dfss2 3907 | . . . . . . . 8 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓)) | |
16 | 1 | elexi 3451 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ V |
17 | vex 3436 | . . . . . . . . . . . 12 ⊢ 𝑎 ∈ V | |
18 | 16, 17 | elimasn 5997 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝑎〉 ∈ 𝑅) |
19 | df-br 5075 | . . . . . . . . . . 11 ⊢ (𝐵𝑅𝑎 ↔ 〈𝐵, 𝑎〉 ∈ 𝑅) | |
20 | 18, 19 | bitr4i 277 | . . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑅 “ {𝐵}) ↔ 𝐵𝑅𝑎) |
21 | 20 | imbi1i 350 | . . . . . . . . 9 ⊢ ((𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓) ↔ (𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) |
22 | 21 | albii 1822 | . . . . . . . 8 ⊢ (∀𝑎(𝑎 ∈ (𝑅 “ {𝐵}) → 𝑎 ∈ 𝑓) ↔ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) |
23 | 15, 22 | bitri 274 | . . . . . . 7 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝑓 ↔ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) |
24 | 14, 23 | anbi12i 627 | . . . . . 6 ⊢ (((𝑅 “ 𝑓) ⊆ 𝑓 ∧ (𝑅 “ {𝐵}) ⊆ 𝑓) ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓))) |
25 | 12, 24 | bitri 274 | . . . . 5 ⊢ ((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 ↔ (𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓))) |
26 | 25 | imbi1i 350 | . . . 4 ⊢ (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ ((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) → 𝐸 ∈ 𝑓)) |
27 | impexp 451 | . . . 4 ⊢ (((𝑅 hereditary 𝑓 ∧ ∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓)) → 𝐸 ∈ 𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓))) | |
28 | 26, 27 | bitri 274 | . . 3 ⊢ (((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ (𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓))) |
29 | 28 | albii 1822 | . 2 ⊢ (∀𝑓((𝑅 “ ({𝐵} ∪ 𝑓)) ⊆ 𝑓 → 𝐸 ∈ 𝑓) ↔ ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓))) |
30 | 5, 7, 29 | 3bitrri 298 | 1 ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 ∈ wcel 2106 {cab 2715 ∪ cun 3885 ⊆ wss 3887 {csn 4561 〈cop 4567 ∩ cint 4879 class class class wbr 5074 “ cima 5592 ‘cfv 6433 t+ctcl 14696 hereditary whe 41380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-seq 13722 df-trcl 14698 df-relexp 14731 df-he 41381 |
This theorem is referenced by: frege77 41548 frege89 41560 |
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