Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opidonOLD Structured version   Visualization version   GIF version

Theorem opidonOLD 37812
Description: Obsolete version of mndpfo 18795 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 4258 . . . 4 (Magma ∩ ExId ) ⊆ Magma
21sseli 4004 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3 opidonOLD.1 . . . . 5 𝑋 = dom dom 𝐺
43ismgmOLD 37810 . . . 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 4259 . . . . 5 (Magma ∩ ExId ) ⊆ ExId
87sseli 4004 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
93isexid 37807 . . . . 5 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
109biimpd 229 . . . 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 3089 . . . . . . 7 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
14 oveq2 7456 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
1614, 15eqeq12d 2756 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
1716rspcv 3631 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18 eqcom 2747 . . . . . . . . . . 11 (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
1914eqeq1d 2742 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
2018, 19bitrid 283 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
2120rspcev 3635 . . . . . . . . 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 3176 . . . . 5 (𝑦𝑋 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2625impcom 407 . . . 4 ((∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦𝑋) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2726ralrimiva 3152 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2811, 27syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
29 foov 7624 . 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 1537  wcel 2108  wral 3067  wrex 3076  cin 3975   × cxp 5698  dom cdm 5700  wf 6569  ontowfo 6571  (class class class)co 7448   ExId cexid 37804  Magmacmagm 37808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-exid 37805  df-mgmOLD 37809
This theorem is referenced by:  rngopidOLD  37813  opidon2OLD  37814
  Copyright terms: Public domain W3C validator