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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 29237 | . . . . 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 266 | . . . 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 1142 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
5 | 1 | eqcomi 2746 | . . 3 ⊢ ran 𝐺 = 𝑋 |
6 | 4, 5 | jctir 521 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) |
7 | dffo2 6755 | . 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) | |
8 | 6, 7 | sylibr 233 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 × cxp 5628 ran crn 5631 ⟶wf 6487 –onto→wfo 6489 (class class class)co 7349 GrpOpcgr 29229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7662 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-fo 6497 df-fv 6499 df-ov 7352 df-grpo 29233 |
This theorem is referenced by: grpocl 29240 grporndm 29250 grporn 29261 nvgf 29358 hhssabloilem 30001 rngosn3 36278 rngodm1dm2 36286 |
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