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Theorem brsi 4761
 Description: Binary relationship over a singleton image. (Contributed by SF, 11-Feb-2015.)
Assertion
Ref Expression
brsi (A SI RBxy(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.
StepHypRef Expression
1 brex 4689 . 2 (A SI RB → (A V B V))
2 snex 4111 . . . . . 6 {x} V
3 snex 4111 . . . . . 6 {y} V
42, 3pm3.2i 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))
75, 6bi2anan9 843 . . . . 5 ((A = {x} B = {y}) → ((A V B V) ↔ ({x} V {y} V)))
84, 7mpbiri 224 . . . 4 ((A = {x} B = {y}) → (A V B V))
983adant3 975 . . 3 ((A = {x} B = {y} xRy) → (A V B V))
109exlimivv 1635 . 2 (xy(A = {x} B = {y} xRy) → (A V B V))
11 eqeq1 2359 . . . . 5 (z = A → (z = {x} ↔ A = {x}))
12113anbi1d 1256 . . . 4 (z = A → ((z = {x} w = {y} xRy) ↔ (A = {x} w = {y} xRy)))
13122exbidv 1628 . . 3 (z = A → (xy(z = {x} w = {y} xRy) ↔ xy(A = {x} w = {y} xRy)))
14 eqeq1 2359 . . . . 5 (w = B → (w = {y} ↔ B = {y}))
15143anbi2d 1257 . . . 4 (w = B → ((A = {x} w = {y} xRy) ↔ (A = {x} B = {y} xRy)))
16152exbidv 1628 . . 3 (w = B → (xy(A = {x} w = {y} xRy) ↔ xy(A = {x} B = {y} xRy)))
17 df-si 4728 . . 3 SI R = {z, w xy(z = {x} w = {y} xRy)}
1813, 16, 17brabg 4706 . 2 ((A V B V) → (A SI RBxy(A = {x} B = {y} xRy)))
191, 10, 18pm5.21nii 342 1 (A SI RBxy(A = {x} B = {y} xRy))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ 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:  cnvsi  5518  dmsi  5519  funsi  5520  brsnsi1  5775  brsnsi2  5776  lecex  6115
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