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Theorem exidreslem 36740
Description: Lemma for exidres 36741 and exidresid 36742. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
exidres.1 𝑋 = ran 𝐺
exidres.2 𝑈 = (GId‘𝐺)
exidres.3 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
exidreslem ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
Distinct variable groups:   𝑥,𝐺   𝑥,𝑌   𝑥,𝑋   𝑥,𝑈   𝑥,𝐻

Proof of Theorem exidreslem
StepHypRef Expression
1 exidres.3 . . . . . . . 8 𝐻 = (𝐺 ↾ (𝑌 × 𝑌))
21dmeqi 5904 . . . . . . 7 dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3 xpss12 5691 . . . . . . . . . . 11 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
43anidms 567 . . . . . . . . . 10 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5 exidres.1 . . . . . . . . . . . . 13 𝑋 = ran 𝐺
65opidon2OLD 36717 . . . . . . . . . . . 12 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
7 fof 6805 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
8 fdm 6726 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
96, 7, 83syl 18 . . . . . . . . . . 11 (𝐺 ∈ (Magma ∩ ExId ) → dom 𝐺 = (𝑋 × 𝑋))
109sseq2d 4014 . . . . . . . . . 10 (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
114, 10imbitrrid 245 . . . . . . . . 9 (𝐺 ∈ (Magma ∩ ExId ) → (𝑌𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
1211imp 407 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13 ssdmres 6004 . . . . . . . 8 ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
1412, 13sylib 217 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
152, 14eqtrid 2784 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom 𝐻 = (𝑌 × 𝑌))
1615dmeqd 5905 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17 dmxpid 5929 . . . . 5 dom (𝑌 × 𝑌) = 𝑌
1816, 17eqtrdi 2788 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = 𝑌)
1918eleq2d 2819 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑈 ∈ dom dom 𝐻𝑈𝑌))
2019biimp3ar 1470 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
21 ssel2 3977 . . . . . . . . . 10 ((𝑌𝑋𝑥𝑌) → 𝑥𝑋)
22 exidres.2 . . . . . . . . . . 11 𝑈 = (GId‘𝐺)
235, 22cmpidelt 36722 . . . . . . . . . 10 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2421, 23sylan2 593 . . . . . . . . 9 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌𝑋𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2524anassrs 468 . . . . . . . 8 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2625adantrl 714 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
271oveqi 7421 . . . . . . . . . . 11 (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28 ovres 7572 . . . . . . . . . . 11 ((𝑈𝑌𝑥𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
2927, 28eqtrid 2784 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
3029eqeq1d 2734 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
311oveqi 7421 . . . . . . . . . . . 12 (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32 ovres 7572 . . . . . . . . . . . 12 ((𝑥𝑌𝑈𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
3331, 32eqtrid 2784 . . . . . . . . . . 11 ((𝑥𝑌𝑈𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3433ancoms 459 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3534eqeq1d 2734 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
3630, 35anbi12d 631 . . . . . . . 8 ((𝑈𝑌𝑥𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3736adantl 482 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3826, 37mpbird 256 . . . . . 6 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
3938anassrs 468 . . . . 5 ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) ∧ 𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4039ralrimiva 3146 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41403impa 1110 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42123adant3 1132 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑌 × 𝑌) ⊆ dom 𝐺)
4342, 13sylib 217 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
442, 43eqtrid 2784 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom 𝐻 = (𝑌 × 𝑌))
4544dmeqd 5905 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
4645, 17eqtrdi 2788 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = 𝑌)
4746raleqdv 3325 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
4841, 47mpbird 256 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4920, 48jca 512 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  cin 3947  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  cres 5678  wf 6539  ontowfo 6541  cfv 6543  (class class class)co 7408  GIdcgi 29738   ExId cexid 36707  Magmacmagm 36711
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7364  df-ov 7411  df-gid 29742  df-exid 36708  df-mgmOLD 36712
This theorem is referenced by:  exidres  36741  exidresid  36742
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