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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege81d | Structured version Visualization version GIF version |
Description: If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 39077. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege81d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege81d.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
frege81d.b | ⊢ (𝜑 → 𝐵 ∈ V) |
frege81d.ab | ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) |
frege81d.he | ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) |
Ref | Expression |
---|---|
frege81d | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege81d.r | . 2 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | frege81d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | elex 3429 | . . 3 ⊢ (𝐴 ∈ 𝑈 → 𝐴 ∈ V) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | frege81d.b | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
6 | frege81d.ab | . 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
7 | frege81d.he | . 2 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) | |
8 | 2 | snssd 4560 | . . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝑈) |
9 | imass2 5746 | . . . 4 ⊢ ({𝐴} ⊆ 𝑈 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) |
11 | 10, 7 | sstrd 3837 | . 2 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) |
12 | 1, 4, 5, 6, 7, 11 | frege77d 38878 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 {csn 4399 class class class wbr 4875 “ cima 5349 ‘cfv 6127 t+ctcl 14110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-seq 13103 df-trcl 14112 df-relexp 14145 |
This theorem is referenced by: frege83d 38880 |
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