Theorem dmsi 5519

Description: Calculate the domain of a singleton image. Theorem X.4.29.I of [Rosser] p. 301. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
dmsi	|- dom SI R = ~P1 dom R

Proof of Theorem dmsi

Dummy variables a b x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anass 938	. . . . . . . 8 |- ((x = {a} /\ y = {b} /\ aRb) <-> (x = {a} /\ (y = {b} /\ aRb)))
2	1	2exbii 1583	. . . . . . 7 |- (E.yE.b(x = {a} /\ y = {b} /\ aRb) <-> E.yE.b(x = {a} /\ (y = {b} /\ aRb)))
3		19.42vv 1907	. . . . . . 7 |- (E.yE.b(x = {a} /\ (y = {b} /\ aRb)) <-> (x = {a} /\ E.yE.b(y = {b} /\ aRb)))
4	2, 3	bitri 240	. . . . . 6 |- (E.yE.b(x = {a} /\ y = {b} /\ aRb) <-> (x = {a} /\ E.yE.b(y = {b} /\ aRb)))
5		snex 4111	. . . . . . . . . . . 12 |- {b} e. _V
6	5	isseti 2865	. . . . . . . . . . 11 |- E.y y = {b}
7		19.41v 1901	. . . . . . . . . . 11 |- (E.y(y = {b} /\ aRb) <-> (E.y y = {b} /\ aRb))
8	6, 7	mpbiran 884	. . . . . . . . . 10 |- (E.y(y = {b} /\ aRb) <-> aRb)
9	8	exbii 1582	. . . . . . . . 9 |- (E.bE.y(y = {b} /\ aRb) <-> E.b aRb)
10		excom 1741	. . . . . . . . 9 |- (E.yE.b(y = {b} /\ aRb) <-> E.bE.y(y = {b} /\ aRb))
11		eldm 4898	. . . . . . . . 9 |- (a e. dom R <-> E.b aRb)
12	9, 10, 11	3bitr4i 268	. . . . . . . 8 |- (E.yE.b(y = {b} /\ aRb) <-> a e. dom R)
13	12	anbi2i 675	. . . . . . 7 |- ((x = {a} /\ E.yE.b(y = {b} /\ aRb)) <-> (x = {a} /\ a e. dom R))
14		ancom 437	. . . . . . 7 |- ((x = {a} /\ a e. dom R) <-> (a e. dom R /\ x = {a}))
15	13, 14	bitri 240	. . . . . 6 |- ((x = {a} /\ E.yE.b(y = {b} /\ aRb)) <-> (a e. dom R /\ x = {a}))
16	4, 15	bitri 240	. . . . 5 |- (E.yE.b(x = {a} /\ y = {b} /\ aRb) <-> (a e. dom R /\ x = {a}))
17	16	exbii 1582	. . . 4 |- (E.aE.yE.b(x = {a} /\ y = {b} /\ aRb) <-> E.a(a e. dom R /\ x = {a}))
18		excom 1741	. . . 4 |- (E.yE.aE.b(x = {a} /\ y = {b} /\ aRb) <-> E.aE.yE.b(x = {a} /\ y = {b} /\ aRb))
19		df-rex 2620	. . . 4 |- (E.a e. dom R x = {a} <-> E.a(a e. dom R /\ x = {a}))
20	17, 18, 19	3bitr4i 268	. . 3 |- (E.yE.aE.b(x = {a} /\ y = {b} /\ aRb) <-> E.a e. dom R x = {a})
21		eldm 4898	. . . 4 |- (x e. dom SI R <-> E.y x SI Ry)
22		brsi 4761	. . . . 5 |- (x SI Ry <-> E.aE.b(x = {a} /\ y = {b} /\ aRb))
23	22	exbii 1582	. . . 4 |- (E.y x SI Ry <-> E.yE.aE.b(x = {a} /\ y = {b} /\ aRb))
24	21, 23	bitri 240	. . 3 |- (x e. dom SI R <-> E.yE.aE.b(x = {a} /\ y = {b} /\ aRb))
25		elpw1 4144	. . 3 |- (x e. ~P1 dom R <-> E.a e. dom R x = {a})
26	20, 24, 25	3bitr4i 268	. 2 |- (x e. dom SI R <-> x e. ~P1 dom R)
27	26	eqriv 2350	1 |- dom SI R = ~P1 dom R

Syntax hints:   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  {csn 3737  ~P₁ cpw1 4135   class class class wbr 4639   SI csi 4720  dom cdm 4772

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ima 4727  df-si 4728  df-cnv 4785  df-rn 4786  df-dm 4787

This theorem is referenced by:  rnsi  5521  enpw1  6062