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Theorem isexid2 36528
Description: If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid2.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isexid2 (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem isexid2
StepHypRef Expression
1 isexid2.1 . 2 𝑋 = ran 𝐺
2 rngopidOLD 36526 . . . . 5 (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
3 elin 3960 . . . . . . 7 (𝐺 ∈ (Magma ∩ ExId ) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ))
4 eqid 2731 . . . . . . . . . . 11 dom dom 𝐺 = dom dom 𝐺
54isexid 36520 . . . . . . . . . 10 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
65ibi 266 . . . . . . . . 9 (𝐺 ∈ ExId → ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
76a1d 25 . . . . . . . 8 (𝐺 ∈ ExId → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
87adantl 482 . . . . . . 7 ((𝐺 ∈ Magma ∧ 𝐺 ∈ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
93, 8sylbi 216 . . . . . 6 (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
10 eqeq2 2743 . . . . . . 7 (ran 𝐺 = dom dom 𝐺 → (𝑋 = ran 𝐺𝑋 = dom dom 𝐺))
11 raleq 3321 . . . . . . . 8 (ran 𝐺 = dom dom 𝐺 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1211rexeqbi1dv 3333 . . . . . . 7 (ran 𝐺 = dom dom 𝐺 → (∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1310, 12imbi12d 344 . . . . . 6 (ran 𝐺 = dom dom 𝐺 → ((𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) ↔ (𝑋 = dom dom 𝐺 → ∃𝑢 ∈ dom dom 𝐺𝑥 ∈ dom dom 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))))
149, 13imbitrrid 245 . . . . 5 (ran 𝐺 = dom dom 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))))
152, 14mpcom 38 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → (𝑋 = ran 𝐺 → ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1615com12 32 . . 3 (𝑋 = ran 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
17 raleq 3321 . . . 4 (𝑋 = ran 𝐺 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1817rexeqbi1dv 3333 . . 3 (𝑋 = ran 𝐺 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
1916, 18sylibrd 258 . 2 (𝑋 = ran 𝐺 → (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
201, 19ax-mp 5 1 (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  cin 3943  dom cdm 5669  ran crn 5670  (class class class)co 7393   ExId cexid 36517  Magmacmagm 36521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-ov 7396  df-exid 36518  df-mgmOLD 36522
This theorem is referenced by:  exidu1  36529
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