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Theorem opidonOLD 33280
Description: Obsolete version of mndpfo 17235 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 3811 . . . 4 (Magma ∩ ExId ) ⊆ Magma
21sseli 3579 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma)
3 opidonOLD.1 . . . . 5 𝑋 = dom dom 𝐺
43ismgmOLD 33278 . . . 4 (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
54ibi 256 . . 3 (𝐺 ∈ Magma → 𝐺:(𝑋 × 𝑋)⟶𝑋)
62, 5syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7 inss2 3812 . . . . 5 (Magma ∩ ExId ) ⊆ ExId
87sseli 3579 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ ExId )
93isexid 33275 . . . . 5 (𝐺 ∈ ExId → (𝐺 ∈ ExId ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
109biimpd 219 . . . 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 473 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
1312ralimi 2947 . . . . . . 7 (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
14 oveq2 6612 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑢𝐺𝑥) = (𝑢𝐺𝑦))
15 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
1614, 15eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑦) = 𝑦))
1716rspcv 3291 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝐺𝑦) = 𝑦))
18 eqcom 2628 . . . . . . . . . . 11 (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑥) = 𝑦)
1914eqeq1d 2623 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑢𝐺𝑥) = 𝑦 ↔ (𝑢𝐺𝑦) = 𝑦))
2018, 19syl5bb 272 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑦 = (𝑢𝐺𝑥) ↔ (𝑢𝐺𝑦) = 𝑦))
2120rspcev 3295 . . . . . . . . 9 ((𝑦𝑋 ∧ (𝑢𝐺𝑦) = 𝑦) → ∃𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2221ex 450 . . . . . . . 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 3010 . . . . 5 (𝑦𝑋 → (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
2625impcom 446 . . . 4 ((∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ 𝑦𝑋) → ∃𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2726ralrimiva 2960 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
2811, 27syl 17 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥))
29 foov 6761 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑦𝑋𝑢𝑋𝑥𝑋 𝑦 = (𝑢𝐺𝑥)))
306, 28, 29sylanbrc 697 1 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554   × cxp 5072  dom cdm 5074  wf 5843  ontowfo 5845  (class class class)co 6604   ExId cexid 33272  Magmacmagm 33276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-ov 6607  df-exid 33273  df-mgmOLD 33277
This theorem is referenced by:  rngopidOLD  33281  opidon2OLD  33282
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