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Theorem grpofo 29239
Description: A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpofo (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)

Proof of Theorem grpofo
Dummy variables 𝑥 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . . . . 6 𝑋 = ran 𝐺
21isgrpo 29237 . . . . 5 (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
32ibi 266 . . . 4 (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
43simp1d 1142 . . 3 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
51eqcomi 2746 . . 3 ran 𝐺 = 𝑋
64, 5jctir 521 . 2 (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋))
7 dffo2 6755 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋))
86, 7sylibr 233 1 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
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  ontowfo 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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