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Theorem iss2 34435
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 3738 . . . . . . . . 9 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2 vex 3343 . . . . . . . . . 10 𝑥 ∈ V
3 vex 3343 . . . . . . . . . 10 𝑦 ∈ V
42, 3opeldm 5483 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
51, 4jca2 557 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
62, 3opelrn 5512 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ ran 𝐴)
71, 6jca2 557 . . . . . . . 8 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
85, 7jcad 556 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴))))
9 anandi 906 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
108, 9syl6ibr 242 . . . . . 6 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
11 df-br 4805 . . . . . . . . 9 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
123ideq 5430 . . . . . . . . 9 (𝑥 I 𝑦𝑥 = 𝑦)
1311, 12bitr3i 266 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
142eldm2 5477 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
15 opeq2 4554 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1615eleq1d 2824 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1716biimprcd 240 . . . . . . . . . . . . . . . 16 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
1813, 17syl5bi 232 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
191, 18sylcom 30 . . . . . . . . . . . . . 14 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2019exlimdv 2010 . . . . . . . . . . . . 13 (𝐴 ⊆ I → (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2114, 20syl5bi 232 . . . . . . . . . . . 12 (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
2216imbi2d 329 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2321, 22syl5ibcom 235 . . . . . . . . . . 11 (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2423imp 444 . . . . . . . . . 10 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2524adantrd 485 . . . . . . . . 9 ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2625ex 449 . . . . . . . 8 (𝐴 ⊆ I → (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2713, 26syl5bi 232 . . . . . . 7 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
2827impd 446 . . . . . 6 (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
2910, 28impbid 202 . . . . 5 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))))
30 opelinxp 34434 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
3130biancom 34320 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴)))
3229, 31syl6bbr 278 . . . 4 (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
3332alrimivv 2005 . . 3 (𝐴 ⊆ I → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
34 reli 5405 . . . . 5 Rel I
35 relss 5363 . . . . 5 (𝐴 ⊆ I → (Rel I → Rel 𝐴))
3634, 35mpi 20 . . . 4 (𝐴 ⊆ I → Rel 𝐴)
37 relinxp 34393 . . . 4 Rel ( I ∩ (dom 𝐴 × ran 𝐴))
38 eqrel 5366 . . . 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 697 . . 3 (𝐴 ⊆ I → (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)))))
4033, 39mpbird 247 . 2 (𝐴 ⊆ I → 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
41 inss1 3976 . . 3 ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I
42 sseq1 3767 . . 3 (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → (𝐴 ⊆ I ↔ ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I ))
4341, 42mpbiri 248 . 2 (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → 𝐴 ⊆ I )
4440, 43impbii 199 1 (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  cin 3714  wss 3715  cop 4327   class class class wbr 4804   I cid 5173   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  cossssid  34540
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