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Theorem exidreslem 37049
Description: Lemma for exidres 37050 and exidresid 37051. (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 566 . . . . . . . . . 10 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5 exidres.1 . . . . . . . . . . . . 13 𝑋 = ran 𝐺
65opidon2OLD 37026 . . . . . . . . . . . 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 406 . . . . . . . 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 2783 . . . . . 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 2787 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = 𝑌)
1918eleq2d 2818 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑈 ∈ dom dom 𝐻𝑈𝑌))
2019biimp3ar 1469 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
21 ssel2 3977 . . . . . . . . . 10 ((𝑌𝑋𝑥𝑌) → 𝑥𝑋)
22 exidres.2 . . . . . . . . . . 11 𝑈 = (GId‘𝐺)
235, 22cmpidelt 37031 . . . . . . . . . 10 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2421, 23sylan2 592 . . . . . . . . 9 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌𝑋𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2524anassrs 467 . . . . . . . 8 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2625adantrl 713 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
271oveqi 7425 . . . . . . . . . . 11 (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28 ovres 7577 . . . . . . . . . . 11 ((𝑈𝑌𝑥𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
2927, 28eqtrid 2783 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
3029eqeq1d 2733 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
311oveqi 7425 . . . . . . . . . . . 12 (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32 ovres 7577 . . . . . . . . . . . 12 ((𝑥𝑌𝑈𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
3331, 32eqtrid 2783 . . . . . . . . . . 11 ((𝑥𝑌𝑈𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3433ancoms 458 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3534eqeq1d 2733 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
3630, 35anbi12d 630 . . . . . . . 8 ((𝑈𝑌𝑥𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3736adantl 481 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3826, 37mpbird 257 . . . . . 6 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
3938anassrs 467 . . . . 5 ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) ∧ 𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4039ralrimiva 3145 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41403impa 1109 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42123adant3 1131 . . . . . . . 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 2783 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom 𝐻 = (𝑌 × 𝑌))
4544dmeqd 5905 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
4645, 17eqtrdi 2787 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = 𝑌)
4746raleqdv 3324 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
4841, 47mpbird 257 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4920, 48jca 511 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 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  cin 3947  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  cres 5678  wf 6539  ontowfo 6541  cfv 6543  (class class class)co 7412  GIdcgi 30011   ExId cexid 37016  Magmacmagm 37020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-gid 30015  df-exid 37017  df-mgmOLD 37021
This theorem is referenced by:  exidres  37050  exidresid  37051
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