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Theorem opidonOLD 34138
Description: Obsolete version of mndpfo 17629 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 4028 . . . 4 (Magma ∩ ExId ) ⊆ Magma
21sseli 3794 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3 opidonOLD.1 . . . . 5 𝑋 = dom dom 𝐺
43ismgmOLD 34136 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
54ibi 259 . . 3 (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 5syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7 inss2 4029 . . . . 5 (Magma ∩ ExId ) ⊆ ExId
87sseli 3794 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
93isexid 34133 . . . . 5 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
109biimpd 221 . . . 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 475 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
1312ralimi 3133 . . . . . . 7 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
14 oveq2 6886 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
1614, 15eqeq12d 2814 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
1716rspcv 3493 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18 eqcom 2806 . . . . . . . . . . 11 (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
1914eqeq1d 2801 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
2018, 19syl5bb 275 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
2120rspcev 3497 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2221ex 402 . . . . . . . 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 3196 . . . . 5 (𝑦𝑋 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2625impcom 397 . . . 4 ((∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦𝑋) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2726ralrimiva 3147 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2811, 27syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
29 foov 7042 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
306, 28, 29sylanbrc 579 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  cin 3768   × cxp 5310  dom cdm 5312  wf 6097  ontowfo 6099  (class class class)co 6878   ExId cexid 34130  Magmacmagm 34134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-ov 6881  df-exid 34131  df-mgmOLD 34135
This theorem is referenced by:  rngopidOLD  34139  opidon2OLD  34140
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