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Mirrors > Home > MPE Home > Th. List > fusgreg2wsplem | Structured version Visualization version GIF version |
Description: Lemma for fusgreg2wsp 30133 and related theorems. (Contributed by AV, 8-Jan-2022.) |
Ref | Expression |
---|---|
frgrhash2wsp.v | ⢠ð = (Vtxâðº) |
fusgreg2wsp.m | ⢠ð = (ð â ð ⊠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) |
Ref | Expression |
---|---|
fusgreg2wsplem | ⢠(ð â ð â (ð â (ðâð) â (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2739 | . . . . 5 ⢠(ð = ð â ((ð€â1) = ð â (ð€â1) = ð)) | |
2 | 1 | rabbidv 3435 | . . . 4 ⢠(ð = ð â {ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð} = {ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) |
3 | fusgreg2wsp.m | . . . 4 ⢠ð = (ð â ð ⊠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) | |
4 | ovex 7447 | . . . . 5 ⢠(2 WSPathsN ðº) â V | |
5 | 4 | rabex 5328 | . . . 4 ⢠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð} â V |
6 | 2, 3, 5 | fvmpt 6999 | . . 3 ⢠(ð â ð â (ðâð) = {ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) |
7 | 6 | eleq2d 2814 | . 2 ⢠(ð â ð â (ð â (ðâð) â ð â {ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð})) |
8 | fveq1 6890 | . . . 4 ⢠(ð€ = ð â (ð€â1) = (ðâ1)) | |
9 | 8 | eqeq1d 2729 | . . 3 ⢠(ð€ = ð â ((ð€â1) = ð â (ðâ1) = ð)) |
10 | 9 | elrab 3680 | . 2 ⢠(ð â {ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð} â (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð)) |
11 | 7, 10 | bitrdi 287 | 1 ⢠(ð â ð â (ð â (ðâð) â (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð))) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 â§ wa 395 = wceq 1534 â wcel 2099 {crab 3427 ⊠cmpt 5225 âcfv 6542 (class class class)co 7414 1c1 11131 2c2 12289 Vtxcvtx 28796 WSPathsN cwwspthsn 29626 |
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-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 |
This theorem is referenced by: fusgr2wsp2nb 30131 fusgreg2wsp 30133 2wspmdisj 30134 |
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