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Mirrors > Home > MPE Home > Th. List > grpofo | Structured version Visualization version GIF version |
Description: A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpfo.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
grpofo | ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfo.1 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
2 | 1 | isgrpo 28578 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))) |
3 | 2 | ibi 270 | . . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))) |
4 | 3 | simp1d 1144 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
5 | 1 | eqcomi 2746 | . . 3 ⊢ ran 𝐺 = 𝑋 |
6 | 4, 5 | jctir 524 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) |
7 | dffo2 6637 | . 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) | |
8 | 6, 7 | sylibr 237 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 × cxp 5549 ran crn 5552 ⟶wf 6376 –onto→wfo 6378 (class class class)co 7213 GrpOpcgr 28570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fo 6386 df-fv 6388 df-ov 7216 df-grpo 28574 |
This theorem is referenced by: grpocl 28581 grporndm 28591 grporn 28602 nvgf 28699 hhssabloilem 29342 rngosn3 35819 rngodm1dm2 35827 |
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