Theorem imainss 4682

Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
imainss	|- (( R "A) i^i B) C_ ( R "(A i^i ( `' R "B)))

Proof of Theorem imainss

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

Step	Hyp	Ref	Expression
1		vex 2554	. . . . . . . . . . 11 |- y e. _V
2		vex 2554	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4461	. . . . . . . . . 10 |- (y `' Rx <-> x Ry)
4		19.8a 1479	. . . . . . . . . 10 |- ((y e. B /\ y `' Rx) -> E.y(y e. B /\ y `' Rx))
5	3, 4	sylan2br 272	. . . . . . . . 9 |- ((y e. B /\ x Ry) -> E.y(y e. B /\ y `' Rx))
6	5	ancoms 255	. . . . . . . 8 |- ((x Ry /\ y e. B) -> E.y(y e. B /\ y `' Rx))
7	6	anim2i 324	. . . . . . 7 |- ((x e. A /\ (x Ry /\ y e. B)) -> (x e. A /\ E.y(y e. B /\ y `' Rx)))
8		simprl 483	. . . . . . 7 |- ((x e. A /\ (x Ry /\ y e. B)) -> x Ry)
9	7, 8	jca 290	. . . . . 6 |- ((x e. A /\ (x Ry /\ y e. B)) -> ((x e. A /\ E.y(y e. B /\ y `' Rx)) /\ x Ry))
10	9	anassrs 380	. . . . 5 |- (((x e. A /\ x Ry) /\ y e. B) -> ((x e. A /\ E.y(y e. B /\ y `' Rx)) /\ x Ry))
11		elin 3120	. . . . . . 7 |- (x e. (A i^i ( `' R "B)) <-> (x e. A /\ x e. ( `' R "B)))
12	2	elima2 4617	. . . . . . . 8 |- (x e. ( `' R "B) <-> E.y(y e. B /\ y `' Rx))
13	12	anbi2i 430	. . . . . . 7 |- ((x e. A /\ x e. ( `' R "B)) <-> (x e. A /\ E.y(y e. B /\ y `' Rx)))
14	11, 13	bitri 173	. . . . . 6 |- (x e. (A i^i ( `' R "B)) <-> (x e. A /\ E.y(y e. B /\ y `' Rx)))
15	14	anbi1i 431	. . . . 5 |- ((x e. (A i^i ( `' R "B)) /\ x Ry) <-> ((x e. A /\ E.y(y e. B /\ y `' Rx)) /\ x Ry))
16	10, 15	sylibr 137	. . . 4 |- (((x e. A /\ x Ry) /\ y e. B) -> (x e. (A i^i ( `' R "B)) /\ x Ry))
17	16	eximi 1488	. . 3 |- (E.x((x e. A /\ x Ry) /\ y e. B) -> E.x(x e. (A i^i ( `' R "B)) /\ x Ry))
18	1	elima2 4617	. . . . 5 |- (y e. ( R "A) <-> E.x(x e. A /\ x Ry))
19	18	anbi1i 431	. . . 4 |- ((y e. ( R "A) /\ y e. B) <-> (E.x(x e. A /\ x Ry) /\ y e. B))
20		elin 3120	. . . 4 |- (y e. (( R "A) i^i B) <-> (y e. ( R "A) /\ y e. B))
21		19.41v 1779	. . . 4 |- (E.x((x e. A /\ x Ry) /\ y e. B) <-> (E.x(x e. A /\ x Ry) /\ y e. B))
22	19, 20, 21	3bitr4i 201	. . 3 |- (y e. (( R "A) i^i B) <-> E.x((x e. A /\ x Ry) /\ y e. B))
23	1	elima2 4617	. . 3 |- (y e. ( R "(A i^i ( `' R "B))) <-> E.x(x e. (A i^i ( `' R "B)) /\ x Ry))
24	17, 22, 23	3imtr4i 190	. 2 |- (y e. (( R "A) i^i B) -> y e. ( R "(A i^i ( `' R "B))))
25	24	ssriv 2943	1 |- (( R "A) i^i B) C_ ( R "(A i^i ( `' R "B)))

Syntax hints:   /\ wa 97  E.wex 1378   e. wcel 1390   i^i cin 2910   C_ wss 2911   class class class wbr 3755   `'ccnv 4287   "cima 4291

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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)