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Theorem grporn 27715
Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grprn.1 𝐺 ∈ GrpOp
grprn.2 dom 𝐺 = (𝑋 × 𝑋)
Assertion
Ref Expression
grporn 𝑋 = ran 𝐺

Proof of Theorem grporn
StepHypRef Expression
1 grprn.1 . . . 4 𝐺 ∈ GrpOp
2 eqid 2771 . . . . 5 ran 𝐺 = ran 𝐺
32grpofo 27693 . . . 4 (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4 fofun 6258 . . . 4 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → Fun 𝐺)
51, 3, 4mp2b 10 . . 3 Fun 𝐺
6 grprn.2 . . 3 dom 𝐺 = (𝑋 × 𝑋)
7 df-fn 6033 . . 3 (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
85, 6, 7mpbir2an 690 . 2 𝐺 Fn (𝑋 × 𝑋)
9 fofn 6259 . . 3 (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺𝐺 Fn (ran 𝐺 × ran 𝐺))
101, 3, 9mp2b 10 . 2 𝐺 Fn (ran 𝐺 × ran 𝐺)
11 fndmu 6131 . . 3 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12 xpid11 5484 . . 3 ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
1311, 12sylib 208 . 2 ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
148, 10, 13mp2an 672 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145   × cxp 5248  dom cdm 5250  ran crn 5251  Fun wfun 6024   Fn wfn 6025  ontowfo 6028  GrpOpcgr 27683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-ov 6799  df-grpo 27687
This theorem is referenced by:  isabloi  27745  isvciOLD  27775  cnidOLD  27777  cnnv  27872  cnnvba  27874  cncph  28014  hilid  28358  hhnv  28362  hhba  28364  hhph  28375  hhssnv  28461
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