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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 8498 | . 2 ⊢ Rel ≼ | |
2 | vex 3444 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | brdom 8504 | . . 3 ⊢ (𝑥 ≼ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1→𝑦) |
4 | vex 3444 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brdom 8504 | . . 3 ⊢ (𝑦 ≼ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1→𝑧) |
6 | exdistrv 1956 | . . . 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 6560 | . . . . . . . 8 ⊢ ((𝑓:𝑦–1-1→𝑧 ∧ 𝑔:𝑥–1-1→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) | |
8 | 7 | ancoms 462 | . . . . . . 7 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1→𝑧) |
9 | vex 3444 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
10 | vex 3444 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
11 | 9, 10 | coex 7617 | . . . . . . . 8 ⊢ (𝑓 ∘ 𝑔) ∈ V |
12 | f1eq1 6544 | . . . . . . . 8 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:𝑥–1-1→𝑧 ↔ (𝑓 ∘ 𝑔):𝑥–1-1→𝑧)) | |
13 | 11, 12 | spcev 3555 | . . . . . . 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 8504 | . . . . . 6 ⊢ (𝑥 ≼ 𝑧 ↔ ∃ℎ ℎ:𝑥–1-1→𝑧) |
16 | 14, 15 | sylibr 237 | . . . . 5 ⊢ ((𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
17 | 16 | exlimivv 1933 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1→𝑦 ∧ 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
18 | 6, 17 | sylbir 238 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1→𝑧) → 𝑥 ≼ 𝑧) |
19 | 3, 5, 18 | syl2anb 600 | . 2 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑧) → 𝑥 ≼ 𝑧) |
20 | 1, 19 | vtoclr 5579 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1781 class class class wbr 5030 ∘ ccom 5523 –1-1→wf1 6321 ≼ cdom 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-dom 8494 |
This theorem is referenced by: endomtr 8550 domentr 8551 cnvct 8569 ssct 8581 undom 8588 sdomdomtr 8634 domsdomtr 8636 xpen 8664 unxpdom2 8710 sucxpdom 8711 fidomdm 8785 hartogs 8992 harword 9011 unxpwdom 9037 harcard 9391 infxpenlem 9424 xpct 9427 indcardi 9452 fodomfi2 9471 infpwfien 9473 inffien 9474 djudoml 9595 djuinf 9599 infdju1 9600 djulepw 9603 unctb 9616 infdjuabs 9617 infdju 9619 infdif 9620 infdif2 9621 infxp 9626 infmap2 9629 fictb 9656 cfslb2n 9679 isfin32i 9776 fin1a2lem12 9822 hsmexlem1 9837 dmct 9935 brdom3 9939 brdom5 9940 brdom4 9941 imadomg 9945 fimact 9946 fnct 9948 mptct 9949 iundomg 9952 uniimadom 9955 ondomon 9974 unirnfdomd 9978 alephval2 9983 iunctb 9985 alephexp1 9990 alephreg 9993 cfpwsdom 9995 gchdomtri 10040 canthnum 10060 canthp1lem1 10063 canthp1 10065 pwfseqlem5 10074 pwxpndom2 10076 pwxpndom 10077 pwdjundom 10078 gchdjuidm 10079 gchxpidm 10080 gchpwdom 10081 gchaclem 10089 gchhar 10090 inar1 10186 rankcf 10188 grudomon 10228 grothac 10241 rpnnen 15572 cctop 21611 1stcfb 22050 2ndcredom 22055 2ndc1stc 22056 1stcrestlem 22057 2ndcctbss 22060 2ndcdisj2 22062 2ndcomap 22063 2ndcsep 22064 dis2ndc 22065 hauspwdom 22106 tx1stc 22255 tx2ndc 22256 met2ndci 23129 opnreen 23436 rectbntr0 23437 uniiccdif 24182 dyadmbl 24204 opnmblALT 24207 mbfimaopnlem 24259 abrexdomjm 30275 mptctf 30479 locfinreflem 31193 sigaclci 31501 omsmeas 31691 sibfof 31708 abrexdom 35168 heiborlem3 35251 ttac 39977 idomsubgmo 40142 pr2dom 40235 tr3dom 40236 uzct 41697 omeiunle 43156 smfaddlem2 43397 smflimlem6 43409 smfmullem4 43426 smfpimbor1lem1 43430 |
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