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Theorem exidreslem 36386
Description: Lemma for exidres 36387 and exidresid 36388. (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 5864 . . . . . . 7 dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3 xpss12 5652 . . . . . . . . . . 11 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
43anidms 568 . . . . . . . . . 10 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5 exidres.1 . . . . . . . . . . . . 13 𝑋 = ran 𝐺
65opidon2OLD 36363 . . . . . . . . . . . 12 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
7 fof 6760 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
8 fdm 6681 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
96, 7, 83syl 18 . . . . . . . . . . 11 (𝐺 ∈ (Magma ∩ ExId ) → dom 𝐺 = (𝑋 × 𝑋))
109sseq2d 3980 . . . . . . . . . 10 (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
114, 10syl5ibr 246 . . . . . . . . 9 (𝐺 ∈ (Magma ∩ ExId ) → (𝑌𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
1211imp 408 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13 ssdmres 5964 . . . . . . . 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 2785 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom 𝐻 = (𝑌 × 𝑌))
1615dmeqd 5865 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17 dmxpid 5889 . . . . 5 dom (𝑌 × 𝑌) = 𝑌
1816, 17eqtrdi 2789 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = 𝑌)
1918eleq2d 2820 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑈 ∈ dom dom 𝐻𝑈𝑌))
2019biimp3ar 1471 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
21 ssel2 3943 . . . . . . . . . 10 ((𝑌𝑋𝑥𝑌) → 𝑥𝑋)
22 exidres.2 . . . . . . . . . . 11 𝑈 = (GId‘𝐺)
235, 22cmpidelt 36368 . . . . . . . . . 10 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2421, 23sylan2 594 . . . . . . . . 9 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌𝑋𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2524anassrs 469 . . . . . . . 8 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2625adantrl 715 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
271oveqi 7374 . . . . . . . . . . 11 (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28 ovres 7524 . . . . . . . . . . 11 ((𝑈𝑌𝑥𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
2927, 28eqtrid 2785 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
3029eqeq1d 2735 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
311oveqi 7374 . . . . . . . . . . . 12 (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32 ovres 7524 . . . . . . . . . . . 12 ((𝑥𝑌𝑈𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
3331, 32eqtrid 2785 . . . . . . . . . . 11 ((𝑥𝑌𝑈𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3433ancoms 460 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3534eqeq1d 2735 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
3630, 35anbi12d 632 . . . . . . . 8 ((𝑈𝑌𝑥𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3736adantl 483 . . . . . . 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 469 . . . . 5 ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) ∧ 𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4039ralrimiva 3140 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41403impa 1111 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42123adant3 1133 . . . . . . . 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 2785 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom 𝐻 = (𝑌 × 𝑌))
4544dmeqd 5865 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
4645, 17eqtrdi 2789 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = 𝑌)
4746raleqdv 3312 . . 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 513 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  cin 3913  wss 3914   × cxp 5635  dom cdm 5637  ran crn 5638  cres 5639  wf 6496  ontowfo 6498  cfv 6500  (class class class)co 7361  GIdcgi 29481   ExId cexid 36353  Magmacmagm 36357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-gid 29485  df-exid 36354  df-mgmOLD 36358
This theorem is referenced by:  exidres  36387  exidresid  36388
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