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Theorem exidu1 36365
Description: Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
exidu1.1 𝑋 = ran 𝐺
Assertion
Ref Expression
exidu1 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem exidu1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 exidu1.1 . . 3 𝑋 = ran 𝐺
21isexid2 36364 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
3 simpl 484 . . . . . . 7 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
43ralimi 3083 . . . . . 6 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
6 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
75, 6eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
87rspcv 3579 . . . . . 6 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
94, 8syl5 34 . . . . 5 (𝑦𝑋 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑦) = 𝑦))
10 simpr 486 . . . . . . 7 (((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑥𝐺𝑦) = 𝑥)
1110ralimi 3083 . . . . . 6 (∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → ∀𝑥𝑋 (𝑥𝐺𝑦) = 𝑥)
12 oveq1 7368 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥𝐺𝑦) = (𝑢𝐺𝑦))
13 id 22 . . . . . . . 8 (𝑥 = 𝑢𝑥 = 𝑢)
1412, 13eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥𝐺𝑦) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑢))
1514rspcv 3579 . . . . . 6 (𝑢𝑋 → (∀𝑥𝑋 (𝑥𝐺𝑦) = 𝑥 → (𝑢𝐺𝑦) = 𝑢))
1611, 15syl5 34 . . . . 5 (𝑢𝑋 → (∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥) → (𝑢𝐺𝑦) = 𝑢))
179, 16im2anan9r 622 . . . 4 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → ((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢)))
18 eqtr2 2757 . . . . 5 (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑦 = 𝑢)
1918equcomd 2023 . . . 4 (((𝑢𝐺𝑦) = 𝑦 ∧ (𝑢𝐺𝑦) = 𝑢) → 𝑢 = 𝑦)
2017, 19syl6 35 . . 3 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦))
2120rgen2 3191 . 2 𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)
22 oveq1 7368 . . . . 5 (𝑢 = 𝑦 → (𝑢𝐺𝑥) = (𝑦𝐺𝑥))
2322eqeq1d 2735 . . . 4 (𝑢 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑦𝐺𝑥) = 𝑥))
2423ovanraleqv 7385 . . 3 (𝑢 = 𝑦 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)))
2524reu4 3693 . 2 (∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑦) = 𝑥)) → 𝑢 = 𝑦)))
262, 21, 25sylanblrc 591 1 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  ∃!wreu 3350  cin 3913  ran crn 5638  (class class class)co 7361   ExId cexid 36353  Magmacmagm 36357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364  df-exid 36354  df-mgmOLD 36358
This theorem is referenced by:  iorlid  36367  cmpidelt  36368  exidresid  36388
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