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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 2733 | . . . . 5 β’ ran πΊ = ran πΊ | |
| 3 | 2 | grpofo 29752 | . . . 4 β’ (πΊ β GrpOp β πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ) |
| 4 | fofun 6807 | . . . 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 6547 | . . 3 β’ (πΊ Fn (π Γ π) β (Fun πΊ β§ dom πΊ = (π Γ π))) | |
| 8 | 5, 6, 7 | mpbir2an 710 | . 2 β’ πΊ Fn (π Γ π) |
| 9 | fofn 6808 | . . 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 6657 | . . 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 5932 | . . 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 691 | 1 β’ π = ran πΊ |
| Colors of variables: wff setvar class |
| Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 Γ cxp 5675 dom cdm 5677 ran crn 5678 Fun wfun 6538 Fn wfn 6539 βontoβwfo 6542 GrpOpcgr 29742 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-ov 7412 df-grpo 29746 |
| This theorem is referenced by: isabloi 29804 isvciOLD 29833 cnidOLD 29835 cnnv 29930 cnnvba 29932 cncph 30072 hilid 30414 hhnv 30418 hhba 30420 hhph 30431 hhssnv 30517 |
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