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| Mirrors > Home > MPE Home > Th. List > domtr | Structured version Visualization version GIF version | ||
| Description: Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
| Ref | Expression |
|---|---|
| domtr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reldom 8991 | . 2 ⊢ Rel ≼ | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 3 | 2 | brdom 9001 | . . 3 ⊢ (𝑥 ≼ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1→𝑦) |
| 4 | vex 3484 | . . . 4 ⊢ 𝑧 ∈ V | |
| 5 | 4 | brdom 9001 | . . 3 ⊢ (𝑦 ≼ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1→𝑧) |
| 6 | exdistrv 1955 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) ↔ (∃𝑔 𝑔:𝑥–1-1→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1→𝑧)) | |
| 7 | f1co 6815 | . . . . . . . 8 ⊢ ((𝑓:𝑦–1-1→𝑧 ∧ 𝑔:𝑥–1-1→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) | |
| 8 | 7 | ancoms 458 | . . . . . . 7 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) |
| 9 | vex 3484 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
| 10 | vex 3484 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
| 11 | 9, 10 | coex 7952 | . . . . . . . 8 ⊢ (𝑓 ∘ 𝑔) ∈ V |
| 12 | f1eq1 6799 | . . . . . . . 8 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:𝑥–1-1→𝑧 ↔ (𝑓 ∘ 𝑔):𝑥–1-1→𝑧)) | |
| 13 | 11, 12 | spcev 3606 | . . . . . . 7 ⊢ ((𝑓 ∘ 𝑔):𝑥–1-1→𝑧 → ∃ℎ ℎ:𝑥–1-1→𝑧) |
| 14 | 8, 13 | syl 17 | . . . . . 6 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → ∃ℎ ℎ:𝑥–1-1→𝑧) |
| 15 | 4 | brdom 9001 | . . . . . 6 ⊢ (𝑥 ≼ 𝑧 ↔ ∃ℎ ℎ:𝑥–1-1→𝑧) |
| 16 | 14, 15 | sylibr 234 | . . . . 5 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
| 17 | 16 | exlimivv 1932 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
| 18 | 6, 17 | sylbir 235 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
| 19 | 3, 5, 18 | syl2anb 598 | . 2 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑧) → 𝑥 ≼ 𝑧) |
| 20 | 1, 19 | vtoclr 5748 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 class class class wbr 5143 ∘ ccom 5689 –1-1→wf1 6558 ≼ cdom 8983 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-dom 8987 |
| This theorem is referenced by: endomtr 9052 domentr 9053 cnvct 9074 ssctOLD 9092 undomOLD 9100 sdomdomtr 9150 domsdomtr 9152 xpen 9180 unxpdom2 9290 sucxpdom 9291 fidomdm 9374 hartogs 9584 harword 9603 unxpwdom 9629 harcard 10018 infxpenlem 10053 xpct 10056 indcardi 10081 fodomfi2 10100 infpwfien 10102 inffien 10103 djudoml 10225 djuinf 10229 infdju1 10230 djulepw 10233 unctb 10244 infdjuabs 10245 infdju 10247 infdif 10248 infdif2 10249 infxp 10254 infmap2 10257 fictb 10284 cfslb2n 10308 isfin32i 10405 fin1a2lem12 10451 hsmexlem1 10466 dmct 10564 brdom3 10568 brdom5 10569 brdom4 10570 imadomg 10574 fimact 10575 fnct 10577 mptct 10578 iundomg 10581 uniimadom 10584 ondomon 10603 unirnfdomd 10607 alephval2 10612 iunctb 10614 alephexp1 10619 alephreg 10622 cfpwsdom 10624 gchdomtri 10669 canthnum 10689 canthp1lem1 10692 canthp1 10694 pwfseqlem5 10703 pwxpndom2 10705 pwxpndom 10706 pwdjundom 10707 gchdjuidm 10708 gchxpidm 10709 gchpwdom 10710 gchaclem 10718 gchhar 10719 inar1 10815 rankcf 10817 grudomon 10857 grothac 10870 rpnnen 16263 cctop 23013 1stcfb 23453 2ndcredom 23458 2ndc1stc 23459 1stcrestlem 23460 2ndcctbss 23463 2ndcdisj2 23465 2ndcomap 23466 2ndcsep 23467 dis2ndc 23468 hauspwdom 23509 tx1stc 23658 tx2ndc 23659 met2ndci 24535 opnreen 24853 rectbntr0 24854 uniiccdif 25613 dyadmbl 25635 opnmblALT 25638 mbfimaopnlem 25690 abrexdomjm 32526 mptctf 32729 locfinreflem 33839 sigaclci 34133 omsmeas 34325 sibfof 34342 abrexdom 37737 heiborlem3 37820 imadomfi 42003 ttac 43048 idomsubgmo 43205 safesnsupfidom1o 43430 pr2dom 43540 tr3dom 43541 uzct 45068 rn1st 45280 omeiunle 46532 smfaddlem2 46779 smflimlem6 46791 smfmullem4 46809 smfpimbor1lem1 46813 |
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