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Theorem exidreslem 35315
Description: Lemma for exidres 35316 and exidresid 35317. (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 5737 . . . . . . 7 dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3 xpss12 5534 . . . . . . . . . . 11 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
43anidms 570 . . . . . . . . . 10 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5 exidres.1 . . . . . . . . . . . . 13 𝑋 = ran 𝐺
65opidon2OLD 35292 . . . . . . . . . . . 12 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
7 fof 6565 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
8 fdm 6495 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
96, 7, 83syl 18 . . . . . . . . . . 11 (𝐺 ∈ (Magma ∩ ExId ) → dom 𝐺 = (𝑋 × 𝑋))
109sseq2d 3947 . . . . . . . . . 10 (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
114, 10syl5ibr 249 . . . . . . . . 9 (𝐺 ∈ (Magma ∩ ExId ) → (𝑌𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
1211imp 410 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13 ssdmres 5841 . . . . . . . 8 ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
1412, 13sylib 221 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
152, 14syl5eq 2845 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom 𝐻 = (𝑌 × 𝑌))
1615dmeqd 5738 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17 dmxpid 5764 . . . . 5 dom (𝑌 × 𝑌) = 𝑌
1816, 17eqtrdi 2849 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = 𝑌)
1918eleq2d 2875 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑈 ∈ dom dom 𝐻𝑈𝑌))
2019biimp3ar 1467 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
21 ssel2 3910 . . . . . . . . . 10 ((𝑌𝑋𝑥𝑌) → 𝑥𝑋)
22 exidres.2 . . . . . . . . . . 11 𝑈 = (GId‘𝐺)
235, 22cmpidelt 35297 . . . . . . . . . 10 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2421, 23sylan2 595 . . . . . . . . 9 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌𝑋𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2524anassrs 471 . . . . . . . 8 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2625adantrl 715 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
271oveqi 7148 . . . . . . . . . . 11 (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28 ovres 7294 . . . . . . . . . . 11 ((𝑈𝑌𝑥𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
2927, 28syl5eq 2845 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
3029eqeq1d 2800 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
311oveqi 7148 . . . . . . . . . . . 12 (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32 ovres 7294 . . . . . . . . . . . 12 ((𝑥𝑌𝑈𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
3331, 32syl5eq 2845 . . . . . . . . . . 11 ((𝑥𝑌𝑈𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3433ancoms 462 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3534eqeq1d 2800 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
3630, 35anbi12d 633 . . . . . . . 8 ((𝑈𝑌𝑥𝑌) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3736adantl 485 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → (((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
3826, 37mpbird 260 . . . . . 6 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
3938anassrs 471 . . . . 5 ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) ∧ 𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4039ralrimiva 3149 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41403impa 1107 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42123adant3 1129 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑌 × 𝑌) ⊆ dom 𝐺)
4342, 13sylib 221 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
442, 43syl5eq 2845 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom 𝐻 = (𝑌 × 𝑌))
4544dmeqd 5738 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
4645, 17eqtrdi 2849 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = 𝑌)
4746raleqdv 3364 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
4841, 47mpbird 260 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4920, 48jca 515 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  cin 3880  wss 3881   × cxp 5517  dom cdm 5519  ran crn 5520  cres 5521  wf 6320  ontowfo 6322  cfv 6324  (class class class)co 7135  GIdcgi 28273   ExId cexid 35282  Magmacmagm 35286
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-riota 7093  df-ov 7138  df-gid 28277  df-exid 35283  df-mgmOLD 35287
This theorem is referenced by:  exidres  35316  exidresid  35317
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