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Theorem rngmgmbs4 34061
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 3201 . . . . 5 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
2 simpl 474 . . . . . . . . 9 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
32eqcomd 2766 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥))
4 oveq2 6822 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥))
54eqeq2d 2770 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑢𝐺𝑦) ↔ 𝑥 = (𝑢𝐺𝑥)))
65rspcev 3449 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑢𝐺𝑥)) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
76ex 449 . . . . . . . 8 (𝑥𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
83, 7syl5 34 . . . . . . 7 (𝑥𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
98reximdv 3154 . . . . . 6 (𝑥𝑋 → (∃𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
109ralimia 3088 . . . . 5 (∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
111, 10syl 17 . . . 4 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
1211anim2i 594 . . 3 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
13 foov 6974 . . 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 6280 . 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 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051   × cxp 5264  ran crn 5267  wf 6045  ontowfo 6047  (class class class)co 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-ov 6817
This theorem is referenced by:  rngorn1eq  34064
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