![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . . . 5 ⊢ ran 𝐺 = ran 𝐺 | |
3 | 2 | grpofo 30527 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺) |
4 | fofun 6821 | . . . 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 6565 | . . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋))) | |
8 | 5, 6, 7 | mpbir2an 711 | . 2 ⊢ 𝐺 Fn (𝑋 × 𝑋) |
9 | fofn 6822 | . . 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 6675 | . . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺)) | |
12 | xpid11 5945 | . . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺) | |
13 | 11, 12 | sylib 218 | . 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺) |
14 | 8, 10, 13 | mp2an 692 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 × cxp 5686 dom cdm 5688 ran crn 5689 Fun wfun 6556 Fn wfn 6557 –onto→wfo 6560 GrpOpcgr 30517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-ov 7433 df-grpo 30521 |
This theorem is referenced by: isabloi 30579 isvciOLD 30608 cnidOLD 30610 cnnv 30705 cnnvba 30707 cncph 30847 hilid 31189 hhnv 31193 hhba 31195 hhph 31206 hhssnv 31292 |
Copyright terms: Public domain | W3C validator |