![]() |
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 29246. (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 â§ 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 9193 | . . 3 ⢠(ð â Fin â (ð â {ð}) â Fin) | |
2 | simpr 484 | . . . . . 6 ⢠((ð â ð ⧠¬ ð â Fin) â ¬ ð â Fin) | |
3 | snfi 9058 | . . . . . 6 ⢠{ð} â Fin | |
4 | difinf 9330 | . . . . . 6 ⢠((¬ ð â Fin â§ {ð} â Fin) â ¬ (ð â {ð}) â Fin) | |
5 | 2, 3, 4 | sylancl 585 | . . . . 5 ⢠((ð â ð ⧠¬ ð â Fin) â ¬ (ð â {ð}) â Fin) |
6 | 5 | ex 412 | . . . 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 5324 | . . . . . . 7 ⢠(ð â {ð}) â V |
13 | mptexg 7227 | . . . . . . 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 2832 | . . . . 5 ⢠(ð â ð â ð¹ â V) |
16 | cusgrfi.p | . . . . . 6 ⢠ð = {ð¥ â ð« ð ⣠âð â ð (ð â ð â§ ð¥ = {ð, ð})} | |
17 | 10, 16, 9 | cusgrfilem2 29244 | . . . . 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 3583 | . . . 4 ⢠(ð â ð â âð ð:(ð â {ð})â1-1-ontoâð) |
20 | bren 8963 | . . . 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 9204 | . . 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 279 | 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 â§ wa 395 = wceq 1534 âwex 1774 â wcel 2099 â wne 2935 âwrex 3065 {crab 3427 Vcvv 3469 â cdif 3941 ð« cpw 4598 {csn 4624 {cpr 4626 class class class wbr 5142 ⊠cmpt 5225 â1-1-ontoâwf1o 6541 âcfv 6542 â cen 8950 Fincfn 8953 Vtxcvtx 28783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-en 8954 df-fin 8957 |
This theorem is referenced by: cusgrfi 29246 |
Copyright terms: Public domain | W3C validator |