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Theorem grpocl 30258
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 30257 . . 3 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
3 fof 6798 . . 3 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
42, 3syl 17 . 2 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
5 fovcdm 7573 . 2 ((𝐺:(𝑋 × 𝑋)⟶𝑋𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
64, 5syl3an1 1160 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098   × cxp 5667  ran crn 5670  wf 6532  ontowfo 6534  (class class class)co 7404  GrpOpcgr 30247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7407  df-grpo 30251
This theorem is referenced by:  grpoidinvlem2  30263  grpoidinvlem3  30264  grpoinvop  30291  grpodivf  30296  grpomuldivass  30299  ablo4  30308  nvgcl  30378  ablo4pnp  37259  ghomco  37270  rngogcl  37291  divrngcl  37336  iscringd  37377
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