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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 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grprn.1 | ⊢ 𝐺 ∈ GrpOp |
grprn.2 | ⊢ dom 𝐺 = (𝑋 × 𝑋) |
Ref | Expression |
---|---|
grporn | ⊢ 𝑋 = ran 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grprn.1 | . . . 4 ⊢ 𝐺 ∈ GrpOp | |
2 | eqid 2738 | . . . . 5 ⊢ ran 𝐺 = ran 𝐺 | |
3 | 2 | grpofo 29270 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺) |
4 | fofun 6755 | . . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → Fun 𝐺) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ Fun 𝐺 |
6 | grprn.2 | . . 3 ⊢ dom 𝐺 = (𝑋 × 𝑋) | |
7 | df-fn 6497 | . . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋))) | |
8 | 5, 6, 7 | mpbir2an 710 | . 2 ⊢ 𝐺 Fn (𝑋 × 𝑋) |
9 | fofn 6756 | . . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺 Fn (ran 𝐺 × ran 𝐺)) | |
10 | 1, 3, 9 | mp2b 10 | . 2 ⊢ 𝐺 Fn (ran 𝐺 × ran 𝐺) |
11 | fndmu 6607 | . . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺)) | |
12 | xpid11 5886 | . . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺) | |
13 | 11, 12 | sylib 217 | . 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺) |
14 | 8, 10, 13 | mp2an 691 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 × cxp 5630 dom cdm 5632 ran crn 5633 Fun wfun 6488 Fn wfn 6489 –onto→wfo 6492 GrpOpcgr 29260 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fo 6500 df-fv 6502 df-ov 7355 df-grpo 29264 |
This theorem is referenced by: isabloi 29322 isvciOLD 29351 cnidOLD 29353 cnnv 29448 cnnvba 29450 cncph 29590 hilid 29932 hhnv 29936 hhba 29938 hhph 29949 hhssnv 30035 |
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