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Theorem iss2 38345
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 3977 . . . . . . . . 9 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
3 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
42, 3opeldm 5918 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
51, 4jca2 513 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
62, 3opelrn 5954 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ ran 𝐴)
71, 6jca2 513 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
85, 7jcad 512 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴))))
9 anandi 676 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
108, 9imbitrrdi 252 . . . . . 6 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
11 df-br 5144 . . . . . . . . 9 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
123ideq 5863 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
1311, 12bitr3i 277 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
142eldm2 5912 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
15 opeq2 4874 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2826 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1716biimprcd 250 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1813, 17biimtrid 242 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
191, 18sylcom 30 . . . . . . . . . . . . . 14 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2019exlimdv 1933 . . . . . . . . . . . . 13 (𝐴 ⊆ I → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2114, 20biimtrid 242 . . . . . . . . . . . 12 (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2216imbi2d 340 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2321, 22syl5ibcom 245 . . . . . . . . . . 11 (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2423imp 406 . . . . . . . . . 10 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2524adantrd 491 . . . . . . . . 9 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2625ex 412 . . . . . . . 8 (𝐴 ⊆ I → (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2713, 26biimtrid 242 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2827impd 410 . . . . . 6 (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2910, 28impbid 212 . . . . 5 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
30 opelinxp 5765 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
3130biancomi 462 . . . . 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 1928 . . 3 (𝐴 ⊆ I → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
34 reli 5836 . . . . 5 Rel I
35 relss 5791 . . . . 5 (𝐴 ⊆ I → (Rel I → Rel 𝐴))
3634, 35mpi 20 . . . 4 (𝐴 ⊆ I → Rel 𝐴)
37 relinxp 5824 . . . 4 Rel ( I ∩ (dom 𝐴 × ran 𝐴))
38 eqrel 5794 . . . 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 586 . . 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 4237 . . 3 ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I
42 sseq1 4009 . . 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 209 1 (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  cin 3950  wss 3951  cop 4632   class class class wbr 5143   I cid 5577   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  cossssid  38468
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