Theorem brsnsi 5773

Description: Binary relationship of singletons in a singleton image. (Contributed by SF, 9-Feb-2015.)

Hypotheses

Ref	Expression
brsnsi.1	⊢ A ∈ V
brsnsi.2	⊢ B ∈ V

Assertion

Ref	Expression
brsnsi	⊢ ({A} SI R{B} ↔ ARB)

Proof of Theorem brsnsi

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

Step	Hyp	Ref	Expression
1		snex 4111	. . 3 ⊢ {A} ∈ V
2		snex 4111	. . 3 ⊢ {B} ∈ V
3		eqeq1 2359	. . . . . 6 ⊢ (z = {A} → (z = {x} ↔ {A} = {x}))
4		eqcom 2355	. . . . . . 7 ⊢ ({A} = {x} ↔ {x} = {A})
5		vex 2862	. . . . . . . 8 ⊢ x ∈ V
6	5	sneqb 3876	. . . . . . 7 ⊢ ({x} = {A} ↔ x = A)
7	4, 6	bitri 240	. . . . . 6 ⊢ ({A} = {x} ↔ x = A)
8	3, 7	syl6bb 252	. . . . 5 ⊢ (z = {A} → (z = {x} ↔ x = A))
9	8	3anbi1d 1256	. . . 4 ⊢ (z = {A} → ((z = {x} ∧ w = {y} ∧ xRy) ↔ (x = A ∧ w = {y} ∧ xRy)))
10	9	2exbidv 1628	. . 3 ⊢ (z = {A} → (∃x∃y(z = {x} ∧ w = {y} ∧ xRy) ↔ ∃x∃y(x = A ∧ w = {y} ∧ xRy)))
11		eqeq1 2359	. . . . . 6 ⊢ (w = {B} → (w = {y} ↔ {B} = {y}))
12		eqcom 2355	. . . . . . 7 ⊢ ({B} = {y} ↔ {y} = {B})
13		vex 2862	. . . . . . . 8 ⊢ y ∈ V
14	13	sneqb 3876	. . . . . . 7 ⊢ ({y} = {B} ↔ y = B)
15	12, 14	bitri 240	. . . . . 6 ⊢ ({B} = {y} ↔ y = B)
16	11, 15	syl6bb 252	. . . . 5 ⊢ (w = {B} → (w = {y} ↔ y = B))
17	16	3anbi2d 1257	. . . 4 ⊢ (w = {B} → ((x = A ∧ w = {y} ∧ xRy) ↔ (x = A ∧ y = B ∧ xRy)))
18	17	2exbidv 1628	. . 3 ⊢ (w = {B} → (∃x∃y(x = A ∧ w = {y} ∧ xRy) ↔ ∃x∃y(x = A ∧ y = B ∧ xRy)))
19		df-si 4728	. . 3 ⊢ SI R = {⟨z, w⟩ ∣ ∃x∃y(z = {x} ∧ w = {y} ∧ xRy)}
20	1, 2, 10, 18, 19	brab 4709	. 2 ⊢ ({A} SI R{B} ↔ ∃x∃y(x = A ∧ y = B ∧ xRy))
21		brsnsi.1	. . 3 ⊢ A ∈ V
22		brsnsi.2	. . 3 ⊢ B ∈ V
23		breq1 4642	. . 3 ⊢ (x = A → (xRy ↔ ARy))
24		breq2 4643	. . 3 ⊢ (y = B → (ARy ↔ ARB))
25	21, 22, 23, 24	ceqsex2v 2896	. 2 ⊢ (∃x∃y(x = A ∧ y = B ∧ xRy) ↔ ARB)
26	20, 25	bitri 240	1 ⊢ ({A} SI R{B} ↔ ARB)

Syntax hints:   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737   class class class wbr 4639   SI csi 4720

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-si 4728

This theorem is referenced by:  opsnelsi  5774  pw1fnex  5852  tcfnex  6244