Theorem erinxp 8580

Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
erinxp.r	⊢ (𝜑 → 𝑅 Er 𝐴)
erinxp.a	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
erinxp	⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Proof of Theorem erinxp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relinxp 5724	. . 3 ⊢ Rel (𝑅 ∩ (𝐵 × 𝐵))
2	1	a1i 11	. 2 ⊢ (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
3		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
4		brinxp2 5664	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	3, 4	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
6	5	simplrd 767	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦 ∈ 𝐵)
7	5	simplld 765	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥 ∈ 𝐵)
8		erinxp.r	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝐴)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
10	5	simprd 496	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
11	9, 10	ersym 8510	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
12		brinxp2 5664	. . 3 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦𝑅𝑥))
13	6, 7, 11, 12	syl21anbrc 1343	. 2 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
14	7	adantrr 714	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥 ∈ 𝐵)
15		simprr 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
16		brinxp2 5664	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑦𝑅𝑧))
17	15, 16	sylib 217	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑦𝑅𝑧))
18	17	simplrd 767	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧 ∈ 𝐵)
19	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
20	10	adantrr 714	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
21	17	simprd 496	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
22	19, 20, 21	ertrd 8514	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
23		brinxp2 5664	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑥𝑅𝑧))
24	14, 18, 22, 23	syl21anbrc 1343	. 2 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
25	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 Er 𝐴)
26		erinxp.a	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
27	26	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
28	25, 27	erref 8518	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥𝑅𝑥)
29	28	ex 413	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥𝑅𝑥))
30	29	pm4.71rd 563	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵)))
31		brin 5126	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥))
32		brxp 5636	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵))
33		anidm 565	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐵)
34	32, 33	bitri 274	. . . . 5 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ 𝑥 ∈ 𝐵)
35	34	anbi2i 623	. . . 4 ⊢ ((𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
36	31, 35	bitri 274	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
37	30, 36	bitr4di 289	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
38	2, 13, 24, 37	iserd 8524	1 ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887   class class class wbr 5074   × cxp 5587  Rel wrel 5594   Er wer 8495

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-er 8498

This theorem is referenced by:  frgpuplem  19378  pi1buni  24203  pi1addf  24210  pi1addval  24211  pi1grplem  24212