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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 43405. (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 | 2 | elexd 3494 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
4 | frege81d.b | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
5 | frege81d.ab | . 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | |
6 | frege81d.he | . 2 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) | |
7 | 2 | snssd 4817 | . . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝑈) |
8 | imass2 6111 | . . . 4 ⊢ ({𝐴} ⊆ 𝑈 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) |
10 | 9, 6 | sstrd 3992 | . 2 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) |
11 | 1, 3, 4, 5, 6, 10 | frege77d 43207 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 {csn 4632 class class class wbr 5152 “ cima 5685 ‘cfv 6553 t+ctcl 14972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14007 df-trcl 14974 df-relexp 15007 |
This theorem is referenced by: frege83d 43209 |
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