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Theorem opidonOLD 35441
 Description: Obsolete version of mndpfo 17946 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 4158 . . . 4 (Magma ∩ ExId ) ⊆ Magma
21sseli 3913 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3 opidonOLD.1 . . . . 5 𝑋 = dom dom 𝐺
43ismgmOLD 35439 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
54ibi 270 . . 3 (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 5syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7 inss2 4159 . . . . 5 (Magma ∩ ExId ) ⊆ ExId
87sseli 3913 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
93isexid 35436 . . . . 5 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
109biimpd 232 . . . 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 486 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
1312ralimi 3128 . . . . . . 7 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
14 oveq2 7153 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
1614, 15eqeq12d 2814 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
1716rspcv 3567 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18 eqcom 2805 . . . . . . . . . . 11 (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
1914eqeq1d 2800 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
2018, 19syl5bb 286 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
2120rspcev 3572 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2221ex 416 . . . . . . . 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 3233 . . . . 5 (𝑦𝑋 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2625impcom 411 . . . 4 ((∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦𝑋) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2726ralrimiva 3149 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2811, 27syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
29 foov 7313 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
306, 28, 29sylanbrc 586 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3882   × cxp 5521  dom cdm 5523  ⟶wf 6328  –onto→wfo 6330  (class class class)co 7145   ExId cexid 35433  Magmacmagm 35437 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fo 6338  df-fv 6340  df-ov 7148  df-exid 35434  df-mgmOLD 35438 This theorem is referenced by:  rngopidOLD  35442  opidon2OLD  35443
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