Theorem brsi 4762

Description: Binary relationship over a singleton image. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
brsi	⊢ (A SI RB ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ xRy))

Distinct variable groups:   x,A,y   x,B,y   x,R,y

Proof of Theorem brsi

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

Step	Hyp	Ref	Expression
1		brex 4690	. 2 ⊢ (A SI RB → (A ∈ V ∧ B ∈ V))
2		snex 4112	. . . . . 6 ⊢ {x} ∈ V
3		snex 4112	. . . . . 6 ⊢ {y} ∈ V
4	2, 3	pm3.2i 441	. . . . 5 ⊢ ({x} ∈ V ∧ {y} ∈ V)
5		eleq1 2413	. . . . . 6 ⊢ (A = {x} → (A ∈ V ↔ {x} ∈ V))
6		eleq1 2413	. . . . . 6 ⊢ (B = {y} → (B ∈ V ↔ {y} ∈ V))
7	5, 6	bi2anan9 843	. . . . 5 ⊢ ((A = {x} ∧ B = {y}) → ((A ∈ V ∧ B ∈ V) ↔ ({x} ∈ V ∧ {y} ∈ V)))
8	4, 7	mpbiri 224	. . . 4 ⊢ ((A = {x} ∧ B = {y}) → (A ∈ V ∧ B ∈ V))
9	8	3adant3 975	. . 3 ⊢ ((A = {x} ∧ B = {y} ∧ xRy) → (A ∈ V ∧ B ∈ V))
10	9	exlimivv 1635	. 2 ⊢ (∃x∃y(A = {x} ∧ B = {y} ∧ xRy) → (A ∈ V ∧ B ∈ V))
11		eqeq1 2359	. . . . 5 ⊢ (z = A → (z = {x} ↔ A = {x}))
12	11	3anbi1d 1256	. . . 4 ⊢ (z = A → ((z = {x} ∧ w = {y} ∧ xRy) ↔ (A = {x} ∧ w = {y} ∧ xRy)))
13	12	2exbidv 1628	. . 3 ⊢ (z = A → (∃x∃y(z = {x} ∧ w = {y} ∧ xRy) ↔ ∃x∃y(A = {x} ∧ w = {y} ∧ xRy)))
14		eqeq1 2359	. . . . 5 ⊢ (w = B → (w = {y} ↔ B = {y}))
15	14	3anbi2d 1257	. . . 4 ⊢ (w = B → ((A = {x} ∧ w = {y} ∧ xRy) ↔ (A = {x} ∧ B = {y} ∧ xRy)))
16	15	2exbidv 1628	. . 3 ⊢ (w = B → (∃x∃y(A = {x} ∧ w = {y} ∧ xRy) ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ xRy)))
17		df-si 4729	. . 3 ⊢ SI R = {⟨z, w⟩ ∣ ∃x∃y(z = {x} ∧ w = {y} ∧ xRy)}
18	13, 16, 17	brabg 4707	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A SI RB ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ xRy)))
19	1, 10, 18	pm5.21nii 342	1 ⊢ (A SI RB ↔ ∃x∃y(A = {x} ∧ B = {y} ∧ xRy))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738   class class class wbr 4640   SI csi 4721

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-si 4729

This theorem is referenced by:  cnvsi  5519  dmsi  5520  funsi  5521  brsnsi1  5776  brsnsi2  5777  lecex  6116