Theorem dmtxp 5802

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 5783	. . . . . 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 2862	. . . . . . . . . 10 ⊢ y ∈ V
10		vex 2862	. . . . . . . . . 10 ⊢ z ∈ V
11	9, 10	opex 4588	. . . . . . . . 9 ⊢ ⟨y, z⟩ ∈ V
12	11	isseti 2865	. . . . . . . 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 4898	. . 3 ⊢ (x ∈ dom (R ⊗ S) ↔ ∃p x(R ⊗ S)p)
19		elin 3219	. . . 4 ⊢ (x ∈ (dom R ∩ dom S) ↔ (x ∈ dom R ∧ x ∈ dom S))
20		eldm 4898	. . . . . 6 ⊢ (x ∈ dom R ↔ ∃y xRy)
21		eldm 4898	. . . . . 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 3208  ⟨cop 4561   class class class wbr 4639  dom cdm 4772   ⊗ ctxp 5735

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-1st 4723  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-2nd 4797  df-txp 5736

This theorem is referenced by:  fntxp  5804