Theorem erth2 5970

Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Hypotheses

Ref	Expression
erth2.1	⊢ (φ → R Er V)
erth2.2	⊢ (φ → dom R = X)
erth2.3	⊢ (φ → A ∈ V)
erth2.4	⊢ (φ → B ∈ X)

Assertion

Ref	Expression
erth2	⊢ (φ → (ARB ↔ [A]R = [B]R))

Proof of Theorem erth2

Step	Hyp	Ref	Expression
1		erth2.1	. . 3 ⊢ (φ → R Er V)
2		erth2.3	. . . 4 ⊢ (φ → A ∈ V)
3		elex 2868	. . . 4 ⊢ (A ∈ V → A ∈ V)
4	2, 3	syl 15	. . 3 ⊢ (φ → A ∈ V)
5		erth2.4	. . . 4 ⊢ (φ → B ∈ X)
6		elex 2868	. . . 4 ⊢ (B ∈ X → B ∈ V)
7	5, 6	syl 15	. . 3 ⊢ (φ → B ∈ V)
8	1, 4, 7	ersymb 5954	. 2 ⊢ (φ → (ARB ↔ BRA))
9		erth2.2	. . . 4 ⊢ (φ → dom R = X)
10	1, 9, 5, 2	erth 5969	. . 3 ⊢ (φ → (BRA ↔ [B]R = [A]R))
11		eqcom 2355	. . 3 ⊢ ([B]R = [A]R ↔ [A]R = [B]R)
12	10, 11	syl6bb 252	. 2 ⊢ (φ → (BRA ↔ [A]R = [B]R))
13	8, 12	bitrd 244	1 ⊢ (φ → (ARB ↔ [A]R = [B]R))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640  dom cdm 4773   Er cer 5899  [cec 5946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948

This theorem is referenced by: (None)