Theorem erinxp 8767

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 5780	. . 3 ⊢ Rel (𝑅 ∩ (𝐵 × 𝐵))
2	1	a1i 11	. 2 ⊢ (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
4		brinxp2 5719	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
5	3, 4	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
6	5	simplrd 769	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦 ∈ 𝐵)
7	5	simplld 767	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥 ∈ 𝐵)
8		erinxp.r	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝐴)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
10	5	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
11	9, 10	ersym 8686	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
12		brinxp2 5719	. . 3 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦𝑅𝑥))
13	6, 7, 11, 12	syl21anbrc 1345	. 2 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
14	7	adantrr 717	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥 ∈ 𝐵)
15		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
16		brinxp2 5719	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑦𝑅𝑧))
17	15, 16	sylib 218	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑦𝑅𝑧))
18	17	simplrd 769	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧 ∈ 𝐵)
19	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
20	10	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
21	17	simprd 495	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
22	19, 20, 21	ertrd 8690	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
23		brinxp2 5719	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑥𝑅𝑧))
24	14, 18, 22, 23	syl21anbrc 1345	. 2 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
25	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 Er 𝐴)
26		erinxp.a	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
27	26	sselda 3949	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
28	25, 27	erref 8694	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥𝑅𝑥)
29	28	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥𝑅𝑥))
30	29	pm4.71rd 562	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵)))
31		brin 5162	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥))
32		brxp 5690	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵))
33		anidm 564	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐵)
34	32, 33	bitri 275	. . . . 5 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ 𝑥 ∈ 𝐵)
35	34	anbi2i 623	. . . 4 ⊢ ((𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
36	31, 35	bitri 275	. . 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 8700	1 ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ∩ cin 3916   ⊆ wss 3917   class class class wbr 5110   × cxp 5639  Rel wrel 5646   Er wer 8671

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-er 8674

This theorem is referenced by:  frgpuplem  19709  pi1buni  24947  pi1addf  24954  pi1addval  24955  pi1grplem  24956