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Theorem opidonOLD 37024
Description: Obsolete version of mndpfo 18683 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
opidonOLD.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
opidonOLD (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)

Proof of Theorem opidonOLD
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4228 . . . 4 (Magma ∩ ExId ) ⊆ Magma
21sseli 3978 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3 opidonOLD.1 . . . . 5 𝑋 = dom dom 𝐺
43ismgmOLD 37022 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
54ibi 267 . . 3 (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 5syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7 inss2 4229 . . . . 5 (Magma ∩ ExId ) ⊆ ExId
87sseli 3978 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
93isexid 37019 . . . . 5 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
109biimpd 228 . . . 4 (𝐺 ∈ ExId → (𝐺 ∈ ExId → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
118, 8, 10sylc 65 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
12 simpl 482 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
1312ralimi 3082 . . . . . . 7 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
14 oveq2 7420 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
1614, 15eqeq12d 2747 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
1716rspcv 3608 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18 eqcom 2738 . . . . . . . . . . 11 (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
1914eqeq1d 2733 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
2018, 19bitrid 283 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
2120rspcev 3612 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2221ex 412 . . . . . . . 8 (𝑦𝑋 → ((𝑢𝐺𝑦) = 𝑦 → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2317, 22syld 47 . . . . . . 7 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2413, 23syl5 34 . . . . . 6 (𝑦𝑋 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2524reximdv 3169 . . . . 5 (𝑦𝑋 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2625impcom 407 . . . 4 ((∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦𝑋) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2726ralrimiva 3145 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2811, 27syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
29 foov 7584 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
306, 28, 29sylanbrc 582 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  cin 3947   × cxp 5674  dom cdm 5676  wf 6539  ontowfo 6541  (class class class)co 7412   ExId cexid 37016  Magmacmagm 37020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7415  df-exid 37017  df-mgmOLD 37021
This theorem is referenced by:  rngopidOLD  37025  opidon2OLD  37026
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