Theorem imainss 4739

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 2560	. . . . . . . . . . 11 |- y e. _V
2		vex 2560	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4518	. . . . . . . . . 10 |- ( y `' R x <-> x R y )
4		19.8a 1482	. . . . . . . . . 10 |- ( ( y e. B /\ y `' R x ) -> E. y ( y e. B /\ y `' R x ) )
5	3, 4	sylan2br 272	. . . . . . . . 9 |- ( ( y e. B /\ x R y ) -> E. y ( y e. B /\ y `' R x ) )
6	5	ancoms 255	. . . . . . . 8 |- ( ( x R y /\ y e. B ) -> E. y ( y e. B /\ y `' R x ) )
7	6	anim2i 324	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
8		simprl 483	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> x R y )
9	7, 8	jca 290	. . . . . 6 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
10	9	anassrs 380	. . . . 5 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
11		elin 3126	. . . . . . 7 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ x e. ( `' R " B ) ) )
12	2	elima2 4674	. . . . . . . 8 |- ( x e. ( `' R " B ) <-> E. y ( y e. B /\ y `' R x ) )
13	12	anbi2i 430	. . . . . . 7 |- ( ( x e. A /\ x e. ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
14	11, 13	bitri 173	. . . . . 6 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
15	14	anbi1i 431	. . . . 5 |- ( ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) <-> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
16	10, 15	sylibr 137	. . . 4 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
17	16	eximi 1491	. . 3 |- ( E. x ( ( x e. A /\ x R y ) /\ y e. B ) -> E. x ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
18	1	elima2 4674	. . . . 5 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
19	18	anbi1i 431	. . . 4 |- ( ( y e. ( R " A ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
20		elin 3126	. . . 4 |- ( y e. ( ( R " A ) i^i B ) <-> ( y e. ( R " A ) /\ y e. B ) )
21		19.41v 1782	. . . 4 |- ( E. x ( ( x e. A /\ x R y ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
22	19, 20, 21	3bitr4i 201	. . 3 |- ( y e. ( ( R " A ) i^i B ) <-> E. x ( ( x e. A /\ x R y ) /\ y e. B ) )
23	1	elima2 4674	. . 3 |- ( y e. ( R " ( A i^i ( `' R " B ) ) ) <-> E. x ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
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 2949	1 |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Syntax hints:   /\ wa 97   E.wex 1381   e. wcel 1393   i^i cin 2916   C_ wss 2917   class class class wbr 3764   `'ccnv 4344   "cima 4348

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)