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Theorem exidreslem 32846
 Description: Lemma for exidres 32847 and exidresid 32848. (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 5247 . . . . . . 7 dom 𝐻 = dom (𝐺 ↾ (𝑌 × 𝑌))
3 xpss12 5148 . . . . . . . . . . 11 ((𝑌𝑋𝑌𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
43anidms 675 . . . . . . . . . 10 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
5 exidres.1 . . . . . . . . . . . . 13 𝑋 = ran 𝐺
65opidon2OLD 32823 . . . . . . . . . . . 12 (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
7 fof 6028 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
8 fdm 5964 . . . . . . . . . . . 12 (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋))
96, 7, 83syl 18 . . . . . . . . . . 11 (𝐺 ∈ (Magma ∩ ExId ) → dom 𝐺 = (𝑋 × 𝑋))
109sseq2d 3596 . . . . . . . . . 10 (𝐺 ∈ (Magma ∩ ExId ) → ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)))
114, 10syl5ibr 235 . . . . . . . . 9 (𝐺 ∈ (Magma ∩ ExId ) → (𝑌𝑋 → (𝑌 × 𝑌) ⊆ dom 𝐺))
1211imp 444 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑌 × 𝑌) ⊆ dom 𝐺)
13 ssdmres 5340 . . . . . . . 8 ((𝑌 × 𝑌) ⊆ dom 𝐺 ↔ dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
1412, 13sylib 207 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
152, 14syl5eq 2656 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom 𝐻 = (𝑌 × 𝑌))
1615dmeqd 5248 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = dom (𝑌 × 𝑌))
17 dmxpid 5266 . . . . 5 dom (𝑌 × 𝑌) = 𝑌
1816, 17syl6eq 2660 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → dom dom 𝐻 = 𝑌)
1918eleq2d 2673 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) → (𝑈 ∈ dom dom 𝐻𝑈𝑌))
2019biimp3ar 1425 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → 𝑈 ∈ dom dom 𝐻)
21 ssel2 3563 . . . . . . . . . 10 ((𝑌𝑋𝑥𝑌) → 𝑥𝑋)
22 exidres.2 . . . . . . . . . . 11 𝑈 = (GId‘𝐺)
235, 22cmpidelt 32828 . . . . . . . . . 10 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2421, 23sylan2 490 . . . . . . . . 9 ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝑌𝑋𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2524anassrs 678 . . . . . . . 8 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
2625adantrl 748 . . . . . . 7 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
271oveqi 6562 . . . . . . . . . . 11 (𝑈𝐻𝑥) = (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥)
28 ovres 6698 . . . . . . . . . . 11 ((𝑈𝑌𝑥𝑌) → (𝑈(𝐺 ↾ (𝑌 × 𝑌))𝑥) = (𝑈𝐺𝑥))
2927, 28syl5eq 2656 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑈𝐻𝑥) = (𝑈𝐺𝑥))
3029eqeq1d 2612 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
311oveqi 6562 . . . . . . . . . . . 12 (𝑥𝐻𝑈) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈)
32 ovres 6698 . . . . . . . . . . . 12 ((𝑥𝑌𝑈𝑌) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑈) = (𝑥𝐺𝑈))
3331, 32syl5eq 2656 . . . . . . . . . . 11 ((𝑥𝑌𝑈𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3433ancoms 468 . . . . . . . . . 10 ((𝑈𝑌𝑥𝑌) → (𝑥𝐻𝑈) = (𝑥𝐺𝑈))
3534eqeq1d 2612 . . . . . . . . 9 ((𝑈𝑌𝑥𝑌) → ((𝑥𝐻𝑈) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
3630, 35anbi12d 743 . . . . . . . 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 246 . . . . . 6 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ (𝑈𝑌𝑥𝑌)) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
3938anassrs 678 . . . . 5 ((((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) ∧ 𝑥𝑌) → ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4039ralrimiva 2949 . . . 4 (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋) ∧ 𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
41403impa 1251 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
42123adant3 1074 . . . . . . . 8 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (𝑌 × 𝑌) ⊆ dom 𝐺)
4342, 13sylib 207 . . . . . . 7 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom (𝐺 ↾ (𝑌 × 𝑌)) = (𝑌 × 𝑌))
442, 43syl5eq 2656 . . . . . 6 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom 𝐻 = (𝑌 × 𝑌))
4544dmeqd 5248 . . . . 5 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = dom (𝑌 × 𝑌))
4645, 17syl6eq 2660 . . . 4 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → dom dom 𝐻 = 𝑌)
4746raleqdv 3121 . . 3 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → (∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥) ↔ ∀𝑥𝑌 ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)))
4841, 47mpbird 246 . 2 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌𝑋𝑈𝑌) → ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))
4920, 48jca 553 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  GIdcgi 26728   ExId cexid 32813  Magmacmagm 32817 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-gid 26732  df-exid 32814  df-mgmOLD 32818 This theorem is referenced by:  exidres  32847  exidresid  32848
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