Theorem iss2 38993

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.

Step	Hyp	Ref	Expression
1		ssel 3931	. . . . . . . . 9 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ I ))
2		vex 3459	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
3		vex 3459	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5897	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	1, 4	jca2 522	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴)))
6	2, 3	opelrn 5933	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
7	1, 6	jca2 522	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
8	5, 7	jcad 521	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴))))
9		anandi 688	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) ↔ ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ∧ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴)))
10	8, 9	imbitrrdi 255	. . . . . 6 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴))))
11		df-br 5110	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
12	3	ideq 5838	. . . . . . . . 9 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
13	11, 12	bitr3i 280	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
14	2	eldm2 5891	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
15		opeq2 4839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
16	15	eleq1d 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
17	16	biimprcd 253	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
18	13, 17	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
19	1, 18	sylcom 31	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
20	19	exlimdv 1963	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ I → (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
21	14, 20	biimtrid 245	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ I → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴))
22	16	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑥⟩ ∈ 𝐴) ↔ (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
23	21, 22	syl5ibcom 248	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
24	23	imp 411	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → (𝑥 ∈ dom 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
25	24	adantrd 496	. . . . . . . . 9 ⊢ ((𝐴 ⊆ I ∧ 𝑥 = 𝑦) → ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
26	25	ex 417	. . . . . . . 8 ⊢ (𝐴 ⊆ I → (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
27	13, 26	biimtrid 245	. . . . . . 7 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ I → ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
28	27	impd 415	. . . . . 6 ⊢ (𝐴 ⊆ I → ((⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ 𝐴))
29	10, 28	impbid 215	. . . . 5 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴))))
30		opelinxp 5741	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
31	30	biancomi 467	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)))
32	29, 31	bitr4di 292	. . . 4 ⊢ (𝐴 ⊆ I → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
33	32	alrimivv 1958	. . 3 ⊢ (𝐴 ⊆ I → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴))))
34		reli 5813	. . . . 5 ⊢ Rel I
35		relss 5768	. . . . 5 ⊢ (𝐴 ⊆ I → (Rel I → Rel 𝐴))
36	34, 35	mpi 21	. . . 4 ⊢ (𝐴 ⊆ I → Rel 𝐴)
37		relinxp 5801	. . . 4 ⊢ Rel ( I ∩ (dom 𝐴 × ran 𝐴))
38		eqrel 5770	. . . 4 ⊢ ((Rel 𝐴 ∧ Rel ( I ∩ (dom 𝐴 × ran 𝐴))) → (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)))))
39	36, 37, 38	sylancl 597	. . 3 ⊢ (𝐴 ⊆ I → (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ∩ (dom 𝐴 × ran 𝐴)))))
40	33, 39	mpbird 260	. 2 ⊢ (𝐴 ⊆ I → 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
41		inss1 4189	. . 3 ⊢ ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I
42		sseq1 3962	. . 3 ⊢ (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → (𝐴 ⊆ I ↔ ( I ∩ (dom 𝐴 × ran 𝐴)) ⊆ I ))
43	41, 42	mpbiri 261	. 2 ⊢ (𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)) → 𝐴 ⊆ I )
44	40, 43	impbii 212	1 ⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905  ⟨cop 4595   class class class wbr 5109   I cid 5555   × cxp 5659  dom cdm 5661  ran crn 5662  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  cossssid  39206