Theorem dmtxp 5803

Description: The domain of a tail cross product is the intersection of the domains of its arguments. (Contributed by SF, 18-Feb-2015.)

Assertion

Ref	Expression
dmtxp	⊢ dom (R ⊗ S) = (dom R ∩ dom S)

Proof of Theorem dmtxp

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

Step	Hyp	Ref	Expression
1		brtxp 5784	. . . . . 6 ⊢ (x(R ⊗ S)p ↔ ∃y∃z(p = ⟨y, z⟩ ∧ xRy ∧ xSz))
2	1	exbii 1582	. . . . 5 ⊢ (∃p x(R ⊗ S)p ↔ ∃p∃y∃z(p = ⟨y, z⟩ ∧ xRy ∧ xSz))
3		exrot3 1744	. . . . 5 ⊢ (∃p∃y∃z(p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ ∃y∃z∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz))
4	2, 3	bitri 240	. . . 4 ⊢ (∃p x(R ⊗ S)p ↔ ∃y∃z∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz))
5		3anass 938	. . . . . . . 8 ⊢ ((p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ (p = ⟨y, z⟩ ∧ (xRy ∧ xSz)))
6	5	exbii 1582	. . . . . . 7 ⊢ (∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ ∃p(p = ⟨y, z⟩ ∧ (xRy ∧ xSz)))
7		19.41v 1901	. . . . . . 7 ⊢ (∃p(p = ⟨y, z⟩ ∧ (xRy ∧ xSz)) ↔ (∃p p = ⟨y, z⟩ ∧ (xRy ∧ xSz)))
8	6, 7	bitri 240	. . . . . 6 ⊢ (∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ (∃p p = ⟨y, z⟩ ∧ (xRy ∧ xSz)))
9		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
10		vex 2863	. . . . . . . . . 10 ⊢ z ∈ V
11	9, 10	opex 4589	. . . . . . . . 9 ⊢ ⟨y, z⟩ ∈ V
12	11	isseti 2866	. . . . . . . 8 ⊢ ∃p p = ⟨y, z⟩
13	12	biantrur 492	. . . . . . 7 ⊢ ((xRy ∧ xSz) ↔ (∃p p = ⟨y, z⟩ ∧ (xRy ∧ xSz)))
14	13	bicomi 193	. . . . . 6 ⊢ ((∃p p = ⟨y, z⟩ ∧ (xRy ∧ xSz)) ↔ (xRy ∧ xSz))
15	8, 14	bitri 240	. . . . 5 ⊢ (∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ (xRy ∧ xSz))
16	15	2exbii 1583	. . . 4 ⊢ (∃y∃z∃p(p = ⟨y, z⟩ ∧ xRy ∧ xSz) ↔ ∃y∃z(xRy ∧ xSz))
17	4, 16	bitri 240	. . 3 ⊢ (∃p x(R ⊗ S)p ↔ ∃y∃z(xRy ∧ xSz))
18		eldm 4899	. . 3 ⊢ (x ∈ dom (R ⊗ S) ↔ ∃p x(R ⊗ S)p)
19		elin 3220	. . . 4 ⊢ (x ∈ (dom R ∩ dom S) ↔ (x ∈ dom R ∧ x ∈ dom S))
20		eldm 4899	. . . . . 6 ⊢ (x ∈ dom R ↔ ∃y xRy)
21		eldm 4899	. . . . . 6 ⊢ (x ∈ dom S ↔ ∃z xSz)
22	20, 21	anbi12i 678	. . . . 5 ⊢ ((x ∈ dom R ∧ x ∈ dom S) ↔ (∃y xRy ∧ ∃z xSz))
23		eeanv 1913	. . . . . 6 ⊢ (∃y∃z(xRy ∧ xSz) ↔ (∃y xRy ∧ ∃z xSz))
24	23	bicomi 193	. . . . 5 ⊢ ((∃y xRy ∧ ∃z xSz) ↔ ∃y∃z(xRy ∧ xSz))
25	22, 24	bitri 240	. . . 4 ⊢ ((x ∈ dom R ∧ x ∈ dom S) ↔ ∃y∃z(xRy ∧ xSz))
26	19, 25	bitri 240	. . 3 ⊢ (x ∈ (dom R ∩ dom S) ↔ ∃y∃z(xRy ∧ xSz))
27	17, 18, 26	3bitr4i 268	. 2 ⊢ (x ∈ dom (R ⊗ S) ↔ x ∈ (dom R ∩ dom S))
28	27	eqriv 2350	1 ⊢ dom (R ⊗ S) = (dom R ∩ dom S)

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640  dom cdm 4773   ⊗ ctxp 5736

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-1st 4724  df-co 4727  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-2nd 4798  df-txp 5737

This theorem is referenced by:  fntxp  5805