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Theorem grpocl 28280
Description: Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpocl ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Proof of Theorem grpocl
StepHypRef Expression
1 grpfo.1 . . . 4 𝑋 = ran 𝐺
21grpofo 28279 . . 3 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
3 fof 6593 . . 3 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
42, 3syl 17 . 2 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
5 fovrn 7321 . 2 ((𝐺:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
64, 5syl3an1 1159 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083   = wceq 1536  wcel 2113   × cxp 5556  ran crn 5559  wf 6354  ontowfo 6356  (class class class)co 7159  GrpOpcgr 28269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-ov 7162  df-grpo 28273
This theorem is referenced by:  grpoidinvlem2  28285  grpoidinvlem3  28286  grpoinvop  28313  grpodivf  28318  grpomuldivass  28321  ablo4  28330  nvgcl  28400  ablo4pnp  35162  ghomco  35173  rngogcl  35194  divrngcl  35239  iscringd  35280
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