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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 2610 | . . . . 5 ⊢ ran 𝐺 = ran 𝐺 | |
3 | 2 | grpofo 26737 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺) |
4 | fofun 6029 | . . . 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 5807 | . . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋))) | |
8 | 5, 6, 7 | mpbir2an 957 | . 2 ⊢ 𝐺 Fn (𝑋 × 𝑋) |
9 | fofn 6030 | . . 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 5906 | . . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺)) | |
12 | xpid11 5268 | . . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺) | |
13 | 11, 12 | sylib 207 | . 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺) |
14 | 8, 10, 13 | mp2an 704 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 × cxp 5036 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 –onto→wfo 5802 GrpOpcgr 26727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 df-grpo 26731 |
This theorem is referenced by: isabloi 26789 isvciOLD 26819 cnidOLD 26821 cnnv 26916 cnnvba 26918 cncph 27058 hilid 27402 hhnv 27406 hhba 27408 hhph 27419 hhssnv 27505 |
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