Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngmgmbs4 Structured version   Visualization version   GIF version

Theorem rngmgmbs4 33362
Description: The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
rngmgmbs4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)
Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem rngmgmbs4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.12 3056 . . . . 5 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
2 simpl 473 . . . . . . . . 9 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
32eqcomd 2627 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥))
4 oveq2 6612 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥))
54eqeq2d 2631 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑢𝐺𝑦) ↔ 𝑥 = (𝑢𝐺𝑥)))
65rspcev 3295 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑢𝐺𝑥)) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
76ex 450 . . . . . . . 8 (𝑥𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
83, 7syl5 34 . . . . . . 7 (𝑥𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
98reximdv 3010 . . . . . 6 (𝑥𝑋 → (∃𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
109ralimia 2945 . . . . 5 (∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
111, 10syl 17 . . . 4 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
1211anim2i 592 . . 3 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
13 foov 6761 . . 3 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
1412, 13sylibr 224 . 2 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
15 forn 6075 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
1614, 15syl 17 1 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908   × cxp 5072  ran crn 5075  wf 5843  ontowfo 5845  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-ov 6607
This theorem is referenced by:  rngorn1eq  33365
  Copyright terms: Public domain W3C validator