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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 43935. (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 3503 | . 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 4807 | . . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝑈) |
8 | imass2 6118 | . . . 4 ⊢ ({𝐴} ⊆ 𝑈 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ (𝑅 “ 𝑈)) |
10 | 9, 6 | sstrd 3993 | . 2 ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) |
11 | 1, 3, 4, 5, 6, 10 | frege77d 43737 | 1 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3479 ⊆ wss 3950 {csn 4624 class class class wbr 5141 “ cima 5686 ‘cfv 6559 t+ctcl 15020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-n0 12523 df-z 12610 df-uz 12875 df-seq 14039 df-trcl 15022 df-relexp 15055 |
This theorem is referenced by: frege83d 43739 |
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