Theorem dminxp 4708

Description: Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
dminxp	⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ∀x ∈ A ∃y ∈ B x𝐶y)

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

Allowed substitution hint:   A(y)

Proof of Theorem dminxp

Step	Hyp	Ref	Expression
1		dfdm4 4470	. . . 4 ⊢ dom (𝐶 ∩ (A × B)) = ran ◡(𝐶 ∩ (A × B))
2		cnvin 4674	. . . . . 6 ⊢ ◡(𝐶 ∩ (A × B)) = (◡𝐶 ∩ ◡(A × B))
3		cnvxp 4685	. . . . . . 7 ⊢ ◡(A × B) = (B × A)
4	3	ineq2i 3129	. . . . . 6 ⊢ (◡𝐶 ∩ ◡(A × B)) = (◡𝐶 ∩ (B × A))
5	2, 4	eqtri 2057	. . . . 5 ⊢ ◡(𝐶 ∩ (A × B)) = (◡𝐶 ∩ (B × A))
6	5	rneqi 4505	. . . 4 ⊢ ran ◡(𝐶 ∩ (A × B)) = ran (◡𝐶 ∩ (B × A))
7	1, 6	eqtri 2057	. . 3 ⊢ dom (𝐶 ∩ (A × B)) = ran (◡𝐶 ∩ (B × A))
8	7	eqeq1i 2044	. 2 ⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ran (◡𝐶 ∩ (B × A)) = A)
9		rninxp 4707	. 2 ⊢ (ran (◡𝐶 ∩ (B × A)) = A ↔ ∀x ∈ A ∃y ∈ B y◡𝐶x)
10		vex 2554	. . . . 5 ⊢ y ∈ V
11		vex 2554	. . . . 5 ⊢ x ∈ V
12	10, 11	brcnv 4461	. . . 4 ⊢ (y◡𝐶x ↔ x𝐶y)
13	12	rexbii 2325	. . 3 ⊢ (∃y ∈ B y◡𝐶x ↔ ∃y ∈ B x𝐶y)
14	13	ralbii 2324	. 2 ⊢ (∀x ∈ A ∃y ∈ B y◡𝐶x ↔ ∀x ∈ A ∃y ∈ B x𝐶y)
15	8, 9, 14	3bitri 195	1 ⊢ (dom (𝐶 ∩ (A × B)) = A ↔ ∀x ∈ A ∃y ∈ B x𝐶y)

Syntax hints:   ↔ wb 98   = wceq 1242  ∀wral 2300  ∃wrex 2301   ∩ cin 2910   class class class wbr 3755   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)