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Theorem iss2 37213
Description: A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.)
Assertion
Ref Expression
iss2 (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))

Proof of Theorem iss2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3976 . . . . . . . . 9 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2 vex 3479 . . . . . . . . . 10 𝑥 ∈ V
3 vex 3479 . . . . . . . . . 10 𝑦 ∈ V
42, 3opeldm 5908 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
51, 4jca2 515 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
62, 3opelrn 5943 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ ran 𝐴)
71, 6jca2 515 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
85, 7jcad 514 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴))))
9 anandi 675 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
108, 9syl6ibr 252 . . . . . 6 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
11 df-br 5150 . . . . . . . . 9 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
123ideq 5853 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
1311, 12bitr3i 277 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
142eldm2 5902 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
15 opeq2 4875 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2819 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1716biimprcd 249 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1813, 17biimtrid 241 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
191, 18sylcom 30 . . . . . . . . . . . . . 14 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2019exlimdv 1937 . . . . . . . . . . . . 13 (𝐴 ⊆ I → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2114, 20biimtrid 241 . . . . . . . . . . . 12 (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2216imbi2d 341 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2321, 22syl5ibcom 244 . . . . . . . . . . 11 (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2423imp 408 . . . . . . . . . 10 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2524adantrd 493 . . . . . . . . 9 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2625ex 414 . . . . . . . 8 (𝐴 ⊆ I → (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2713, 26biimtrid 241 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2827impd 412 . . . . . 6 (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2910, 28impbid 211 . . . . 5 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
30 opelinxp 5756 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
3130biancomi 464 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)))
3229, 31bitr4di 289 . . . 4 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
3332alrimivv 1932 . . 3 (𝐴 ⊆ I → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
34 reli 5827 . . . . 5 Rel I
35 relss 5782 . . . . 5 (𝐴 ⊆ I → (Rel I → Rel 𝐴))
3634, 35mpi 20 . . . 4 (𝐴 ⊆ I → Rel 𝐴)
37 relinxp 5815 . . . 4 Rel ( I ∩ (dom 𝐴 × ran 𝐴))
38 eqrel 5785 . . . 4 ((Rel 𝐴 ∧ Rel ( I ∩ (dom 𝐴 × ran 𝐴))) → (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)))))
3936, 37, 38sylancl 587 . . 3 (𝐴 ⊆ I → (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)))))
4033, 39mpbird 257 . 2 (𝐴 ⊆ I → 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
41 inss1 4229 . . 3 ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I
42 sseq1 4008 . . 3 (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → (𝐴 ⊆ I ↔ ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I ))
4341, 42mpbiri 258 . 2 (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → 𝐴 ⊆ I )
4440, 43impbii 208 1 (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  cin 3948  wss 3949  cop 4635   class class class wbr 5149   I cid 5574   × cxp 5675  dom cdm 5677  ran crn 5678  Rel wrel 5682
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  cossssid  37337
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