Theorem erth 5969

Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

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

Assertion

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

Proof of Theorem erth

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erth.1	. . . . . . . 8 ⊢ (φ → R Er V)
2	1	adantr 451	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → R Er V)
3		erth.4	. . . . . . . . 9 ⊢ (φ → B ∈ V)
4		elex 2868	. . . . . . . . 9 ⊢ (B ∈ V → B ∈ V)
5	3, 4	syl 15	. . . . . . . 8 ⊢ (φ → B ∈ V)
6	5	adantr 451	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → B ∈ V)
7		erth.3	. . . . . . . . 9 ⊢ (φ → A ∈ X)
8		elex 2868	. . . . . . . . 9 ⊢ (A ∈ X → A ∈ V)
9	7, 8	syl 15	. . . . . . . 8 ⊢ (φ → A ∈ V)
10	9	adantr 451	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → A ∈ V)
11		vex 2863	. . . . . . . 8 ⊢ x ∈ V
12	11	a1i 10	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → x ∈ V)
13		simprl 732	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → ARB)
14		simprr 733	. . . . . . 7 ⊢ ((φ ∧ (ARB ∧ ARx)) → ARx)
15	2, 6, 10, 12, 13, 14	ertr3d 5958	. . . . . 6 ⊢ ((φ ∧ (ARB ∧ ARx)) → BRx)
16	15	expr 598	. . . . 5 ⊢ ((φ ∧ ARB) → (ARx → BRx))
17	11	a1i 10	. . . . . . 7 ⊢ (φ → x ∈ V)
18	1, 9, 5, 17	ertr 5955	. . . . . 6 ⊢ (φ → ((ARB ∧ BRx) → ARx))
19	18	expdimp 426	. . . . 5 ⊢ ((φ ∧ ARB) → (BRx → ARx))
20	16, 19	impbid 183	. . . 4 ⊢ ((φ ∧ ARB) → (ARx ↔ BRx))
21	20	abbidv 2468	. . 3 ⊢ ((φ ∧ ARB) → {x ∣ ARx} = {x ∣ BRx})
22		dfec2 5949	. . 3 ⊢ [A]R = {x ∣ ARx}
23		dfec2 5949	. . 3 ⊢ [B]R = {x ∣ BRx}
24	21, 22, 23	3eqtr4g 2410	. 2 ⊢ ((φ ∧ ARB) → [A]R = [B]R)
25	1	adantr 451	. . 3 ⊢ ((φ ∧ [A]R = [B]R) → R Er V)
26		simpl 443	. . . 4 ⊢ ((φ ∧ [A]R = [B]R) → φ)
27	26, 3, 4	3syl 18	. . 3 ⊢ ((φ ∧ [A]R = [B]R) → B ∈ V)
28	26, 7, 8	3syl 18	. . 3 ⊢ ((φ ∧ [A]R = [B]R) → A ∈ V)
29		erth.2	. . . . . . . 8 ⊢ (φ → dom R = X)
30	1, 29, 7	erref 5960	. . . . . . 7 ⊢ (φ → ARA)
31		elec 5965	. . . . . . 7 ⊢ (A ∈ [A]R ↔ ARA)
32	30, 31	sylibr 203	. . . . . 6 ⊢ (φ → A ∈ [A]R)
33		eleq2 2414	. . . . . 6 ⊢ ([A]R = [B]R → (A ∈ [A]R ↔ A ∈ [B]R))
34	32, 33	syl5ibcom 211	. . . . 5 ⊢ (φ → ([A]R = [B]R → A ∈ [B]R))
35	34	imp 418	. . . 4 ⊢ ((φ ∧ [A]R = [B]R) → A ∈ [B]R)
36		elec 5965	. . . 4 ⊢ (A ∈ [B]R ↔ BRA)
37	35, 36	sylib 188	. . 3 ⊢ ((φ ∧ [A]R = [B]R) → BRA)
38	25, 27, 28, 37	ersym 5953	. 2 ⊢ ((φ ∧ [A]R = [B]R) → ARB)
39	24, 38	impbida 805	1 ⊢ (φ → (ARB ↔ [A]R = [B]R))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  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:  erth2  5970  erthi  5971  eqncg  6127  ncseqnc  6129