Theorem brab1 4685

Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Assertion

Ref	Expression
brab1	⊢ (xRA ↔ x ∈ {z ∣ zRA})

Distinct variable groups:   z,A   z,R

Allowed substitution hints:   A(x)   R(x)

Proof of Theorem brab1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . 3 ⊢ x ∈ V
2		breq1 4643	. . . 4 ⊢ (z = y → (zRA ↔ yRA))
3		breq1 4643	. . . 4 ⊢ (y = x → (yRA ↔ xRA))
4	2, 3	sbcie2g 3080	. . 3 ⊢ (x ∈ V → ([̣x / z]̣zRA ↔ xRA))
5	1, 4	ax-mp 5	. 2 ⊢ ([̣x / z]̣zRA ↔ xRA)
6		df-sbc 3048	. 2 ⊢ ([̣x / z]̣zRA ↔ x ∈ {z ∣ zRA})
7	5, 6	bitr3i 242	1 ⊢ (xRA ↔ x ∈ {z ∣ zRA})

Syntax hints:   ↔ wb 176   ∈ wcel 1710  {cab 2339  Vcvv 2860  [̣wsbc 3047   class class class wbr 4640

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641

This theorem is referenced by: (None)