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| Mirrors > Home > MPE Home > Th. List > grporn | Structured version Visualization version GIF version | ||
| Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form π = ran πΊ. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grprn.1 | β’ πΊ β GrpOp |
| grprn.2 | β’ dom πΊ = (π Γ π) |
| Ref | Expression |
|---|---|
| grporn | β’ π = ran πΊ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grprn.1 | . . . 4 β’ πΊ β GrpOp | |
| 2 | eqid 2732 | . . . . 5 β’ ran πΊ = ran πΊ | |
| 3 | 2 | grpofo 30007 | . . . 4 β’ (πΊ β GrpOp β πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ) |
| 4 | fofun 6806 | . . . 4 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β Fun πΊ) | |
| 5 | 1, 3, 4 | mp2b 10 | . . 3 β’ Fun πΊ |
| 6 | grprn.2 | . . 3 β’ dom πΊ = (π Γ π) | |
| 7 | df-fn 6546 | . . 3 β’ (πΊ Fn (π Γ π) β (Fun πΊ β§ dom πΊ = (π Γ π))) | |
| 8 | 5, 6, 7 | mpbir2an 709 | . 2 β’ πΊ Fn (π Γ π) |
| 9 | fofn 6807 | . . 3 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β πΊ Fn (ran πΊ Γ ran πΊ)) | |
| 10 | 1, 3, 9 | mp2b 10 | . 2 β’ πΊ Fn (ran πΊ Γ ran πΊ) |
| 11 | fndmu 6656 | . . 3 β’ ((πΊ Fn (π Γ π) β§ πΊ Fn (ran πΊ Γ ran πΊ)) β (π Γ π) = (ran πΊ Γ ran πΊ)) | |
| 12 | xpid11 5931 | . . 3 β’ ((π Γ π) = (ran πΊ Γ ran πΊ) β π = ran πΊ) | |
| 13 | 11, 12 | sylib 217 | . 2 β’ ((πΊ Fn (π Γ π) β§ πΊ Fn (ran πΊ Γ ran πΊ)) β π = ran πΊ) |
| 14 | 8, 10, 13 | mp2an 690 | 1 β’ π = ran πΊ |
| Colors of variables: wff setvar class |
| Syntax hints: β§ wa 396 = wceq 1541 β wcel 2106 Γ cxp 5674 dom cdm 5676 ran crn 5677 Fun wfun 6537 Fn wfn 6538 βontoβwfo 6541 GrpOpcgr 29997 |
| 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-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 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-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-ov 7414 df-grpo 30001 |
| This theorem is referenced by: isabloi 30059 isvciOLD 30088 cnidOLD 30090 cnnv 30185 cnnvba 30187 cncph 30327 hilid 30669 hhnv 30673 hhba 30675 hhph 30686 hhssnv 30772 |
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