Theorem imainss 2649

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

Assertion

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

Proof of Theorem imainss

Step	Hyp	Ref	Expression
1		19.8a 712	. . . . . . . . . 10 |- ((y e. B /\ y`'Rx) -> E.y(y e. B /\ y`'Rx))
2		visset 1350	. . . . . . . . . . 11 |- y e. V
3		visset 1350	. . . . . . . . . . 11 |- x e. V
4	2, 3	brcnv 2519	. . . . . . . . . 10 |- (y`'Rx <-> xRy)
5	1, 4	sylan2br 348	. . . . . . . . 9 |- ((y e. B /\ xRy) -> E.y(y e. B /\ y`'Rx))
6	5	ancoms 334	. . . . . . . 8 |- ((xRy /\ y e. B) -> E.y(y e. B /\ y`'Rx))
7	6	anim2i 270	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> (x e. A /\ E.y(y e. B /\ y`'Rx)))
8		pm3.26 256	. . . . . . . 8 |- ((xRy /\ y e. B) -> xRy)
9	8	adantl 305	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> xRy)
10	7, 9	jca 236	. . . . . 6 |- ((x e. A /\ (xRy /\ y e. B)) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
11	10	anassrs 338	. . . . 5 |- (((x e. A /\ xRy) /\ y e. B) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
12		elin 1635	. . . . . . 7 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ x e. (`'R"B)))
13	3	elima2 2607	. . . . . . . 8 |- (x e. (`'R"B) <-> E.y(y e. B /\ y`'Rx))
14	13	anbi2i 367	. . . . . . 7 |- ((x e. A /\ x e. (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
15	12, 14	bitr 151	. . . . . 6 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
16	15	anbi1i 368	. . . . 5 |- ((x e. (A i^i (`'R"B)) /\ xRy) <-> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
17	11, 16	sylibr 175	. . . 4 |- (((x e. A /\ xRy) /\ y e. B) -> (x e. (A i^i (`'R"B)) /\ xRy))
18	17	19.22i 723	. . 3 |- (E.x((x e. A /\ xRy) /\ y e. B) -> E.x(x e. (A i^i (`'R"B)) /\ xRy))
19	2	elima2 2607	. . . . 5 |- (y e. (R"A) <-> E.x(x e. A /\ xRy))
20	19	anbi1i 368	. . . 4 |- ((y e. (R"A) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
21		elin 1635	. . . 4 |- (y e. ((R"A) i^i B) <-> (y e. (R"A) /\ y e. B))
22		19.41v 963	. . . 4 |- (E.x((x e. A /\ xRy) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
23	20, 21, 22	3bitr4 158	. . 3 |- (y e. ((R"A) i^i B) <-> E.x((x e. A /\ xRy) /\ y e. B))
24	2	elima2 2607	. . 3 |- (y e. (R"(A i^i (`'R"B))) <-> E.x(x e. (A i^i (`'R"B)) /\ xRy))
25	18, 23, 24	3imtr4 192	. 2 |- (y e. ((R"A) i^i B) -> y e. (R"(A i^i (`'R"B))))
26	25	ssriv 1508	1 |- ((R"A) i^i B) (_ (R"(A i^i (`'R"B)))

Syntax hints:   /\ wa 196  E.wex 678   e. wcel 1092   i^i cin 1486   (_ wss 1487   class class class wbr 2054  `'ccnv 2409  "cima 2413

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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