Theorem imainss 4331

Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66.

Assertion

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

Proof of Theorem imainss

Step	Hyp	Ref	Expression
1		19.8a 1376	. . . . . . . . . 10 |- ((y e. B /\ y`'Rx) -> E.y(y e. B /\ y`'Rx))
2		visset 2295	. . . . . . . . . . 11 |- y e. _V
3		visset 2295	. . . . . . . . . . 11 |- x e. _V
4	2, 3	brcnv 4144	. . . . . . . . . 10 |- (y`'Rx <-> xRy)
5	1, 4	sylan2br 502	. . . . . . . . 9 |- ((y e. B /\ xRy) -> E.y(y e. B /\ y`'Rx))
6	5	ancoms 484	. . . . . . . 8 |- ((xRy /\ y e. B) -> E.y(y e. B /\ y`'Rx))
7	6	anim2i 362	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> (x e. A /\ E.y(y e. B /\ y`'Rx)))
8		simprl 450	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> xRy)
9	7, 8	jca 310	. . . . . 6 |- ((x e. A /\ (xRy /\ y e. B)) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
10	9	anassrs 489	. . . . 5 |- (((x e. A /\ xRy) /\ y e. B) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
11		elin 2786	. . . . . . 7 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ x e. (`'R"B)))
12	3	elima2 4271	. . . . . . . 8 |- (x e. (`'R"B) <-> E.y(y e. B /\ y`'Rx))
13	12	anbi2i 538	. . . . . . 7 |- ((x e. A /\ x e. (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
14	11, 13	bitri 190	. . . . . 6 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
15	14	anbi1i 539	. . . . 5 |- ((x e. (A i^i (`'R"B)) /\ xRy) <-> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
16	10, 15	sylibr 217	. . . 4 |- (((x e. A /\ xRy) /\ y e. B) -> (x e. (A i^i (`'R"B)) /\ xRy))
17	16	eximi 1387	. . 3 |- (E.x((x e. A /\ xRy) /\ y e. B) -> E.x(x e. (A i^i (`'R"B)) /\ xRy))
18	2	elima2 4271	. . . . 5 |- (y e. (R"A) <-> E.x(x e. A /\ xRy))
19	18	anbi1i 539	. . . 4 |- ((y e. (R"A) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
20		elin 2786	. . . 4 |- (y e. ((R"A) i^i B) <-> (y e. (R"A) /\ y e. B))
21		19.41v 1685	. . . 4 |- (E.x((x e. A /\ xRy) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
22	19, 20, 21	3bitr4i 200	. . 3 |- (y e. ((R"A) i^i B) <-> E.x((x e. A /\ xRy) /\ y e. B))
23	2	elima2 4271	. . 3 |- (y e. (R"(A i^i (`'R"B))) <-> E.x(x e. (A i^i (`'R"B)) /\ xRy))
24	17, 22, 23	3imtr4i 236	. 2 |- (y e. ((R"A) i^i B) -> y e. (R"(A i^i (`'R"B))))
25	24	ssriv 2621	1 |- ((R"A) i^i B) C_ (R"(A i^i (`'R"B)))

Syntax hints:   /\ wa 240   e. wcel 1300  E.wex 1326   i^i cin 2592   C_ wss 2593   class class class wbr 3338  `'ccnv 3985  "cima 3989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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