![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cusgrfilem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for cusgrfi 28712. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
Ref | Expression |
---|---|
cusgrfi.v | ⢠ð = (Vtxâðº) |
cusgrfi.p | ⢠ð = {ð¥ â ð« ð ⣠âð â ð (ð â ð ⧠ð¥ = {ð, ð})} |
cusgrfi.f | ⢠ð¹ = (ð¥ â (ð â {ð}) ⊠{ð¥, ð}) |
Ref | Expression |
---|---|
cusgrfilem3 | ⢠(ð â ð â (ð â Fin â ð â Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diffi 9178 | . . 3 ⢠(ð â Fin â (ð â {ð}) â Fin) | |
2 | simpr 485 | . . . . . 6 ⢠((ð â ð ⧠¬ ð â Fin) â ¬ ð â Fin) | |
3 | snfi 9043 | . . . . . 6 ⢠{ð} â Fin | |
4 | difinf 9315 | . . . . . 6 ⢠((¬ ð â Fin ⧠{ð} â Fin) â ¬ (ð â {ð}) â Fin) | |
5 | 2, 3, 4 | sylancl 586 | . . . . 5 ⢠((ð â ð ⧠¬ ð â Fin) â ¬ (ð â {ð}) â Fin) |
6 | 5 | ex 413 | . . . 4 ⢠(ð â ð â (¬ ð â Fin â ¬ (ð â {ð}) â Fin)) |
7 | 6 | con4d 115 | . . 3 ⢠(ð â ð â ((ð â {ð}) â Fin â ð â Fin)) |
8 | 1, 7 | impbid2 225 | . 2 ⢠(ð â ð â (ð â Fin â (ð â {ð}) â Fin)) |
9 | cusgrfi.f | . . . . . 6 ⢠ð¹ = (ð¥ â (ð â {ð}) ⊠{ð¥, ð}) | |
10 | cusgrfi.v | . . . . . . . . 9 ⢠ð = (Vtxâðº) | |
11 | 10 | fvexi 6905 | . . . . . . . 8 ⢠ð â V |
12 | 11 | difexi 5328 | . . . . . . 7 ⢠(ð â {ð}) â V |
13 | mptexg 7222 | . . . . . . 7 ⢠((ð â {ð}) â V â (ð¥ â (ð â {ð}) ⊠{ð¥, ð}) â V) | |
14 | 12, 13 | mp1i 13 | . . . . . 6 ⢠(ð â ð â (ð¥ â (ð â {ð}) ⊠{ð¥, ð}) â V) |
15 | 9, 14 | eqeltrid 2837 | . . . . 5 ⢠(ð â ð â ð¹ â V) |
16 | cusgrfi.p | . . . . . 6 ⢠ð = {ð¥ â ð« ð ⣠âð â ð (ð â ð ⧠ð¥ = {ð, ð})} | |
17 | 10, 16, 9 | cusgrfilem2 28710 | . . . . 5 ⢠(ð â ð â ð¹:(ð â {ð})â1-1-ontoâð) |
18 | f1oeq1 6821 | . . . . 5 ⢠(ð = ð¹ â (ð:(ð â {ð})â1-1-ontoâð â ð¹:(ð â {ð})â1-1-ontoâð)) | |
19 | 15, 17, 18 | spcedv 3588 | . . . 4 ⢠(ð â ð â âð ð:(ð â {ð})â1-1-ontoâð) |
20 | bren 8948 | . . . 4 ⢠((ð â {ð}) â ð â âð ð:(ð â {ð})â1-1-ontoâð) | |
21 | 19, 20 | sylibr 233 | . . 3 ⢠(ð â ð â (ð â {ð}) â ð) |
22 | enfi 9189 | . . 3 ⢠((ð â {ð}) â ð â ((ð â {ð}) â Fin â ð â Fin)) | |
23 | 21, 22 | syl 17 | . 2 ⢠(ð â ð â ((ð â {ð}) â Fin â ð â Fin)) |
24 | 8, 23 | bitrd 278 | 1 ⢠(ð â ð â (ð â Fin â ð â Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 â wi 4 â wb 205 ⧠wa 396 = wceq 1541 âwex 1781 â wcel 2106 â wne 2940 âwrex 3070 {crab 3432 Vcvv 3474 â cdif 3945 ð« cpw 4602 {csn 4628 {cpr 4630 class class class wbr 5148 ⊠cmpt 5231 â1-1-ontoâwf1o 6542 âcfv 6543 â cen 8935 Fincfn 8938 Vtxcvtx 28253 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7855 df-1o 8465 df-en 8939 df-fin 8942 |
This theorem is referenced by: cusgrfi 28712 |
Copyright terms: Public domain | W3C validator |