Theorem imainss 3463

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 1029	. . . . . . . . . 10 |- ((y e. B /\ y`'Rx) -> E.y(y e. B /\ y`'Rx))
2		visset 1813	. . . . . . . . . . 11 |- y e. V
3		visset 1813	. . . . . . . . . . 11 |- x e. V
4	2, 3	brcnv 3299	. . . . . . . . . 10 |- (y`'Rx <-> xRy)
5	1, 4	sylan2br 453	. . . . . . . . 9 |- ((y e. B /\ xRy) -> E.y(y e. B /\ y`'Rx))
6	5	ancoms 436	. . . . . . . 8 |- ((xRy /\ y e. B) -> E.y(y e. B /\ y`'Rx))
7	6	anim2i 335	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> (x e. A /\ E.y(y e. B /\ y`'Rx)))
8		simprl 414	. . . . . . 7 |- ((x e. A /\ (xRy /\ y e. B)) -> xRy)
9	7, 8	jca 288	. . . . . 6 |- ((x e. A /\ (xRy /\ y e. B)) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
10	9	anassrs 441	. . . . 5 |- (((x e. A /\ xRy) /\ y e. B) -> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
11		elin 2207	. . . . . . 7 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ x e. (`'R"B)))
12	3	elima2 3409	. . . . . . . 8 |- (x e. (`'R"B) <-> E.y(y e. B /\ y`'Rx))
13	12	anbi2i 480	. . . . . . 7 |- ((x e. A /\ x e. (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
14	11, 13	bitr 173	. . . . . 6 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ E.y(y e. B /\ y`'Rx)))
15	14	anbi1i 481	. . . . 5 |- ((x e. (A i^i (`'R"B)) /\ xRy) <-> ((x e. A /\ E.y(y e. B /\ y`'Rx)) /\ xRy))
16	10, 15	sylibr 200	. . . 4 |- (((x e. A /\ xRy) /\ y e. B) -> (x e. (A i^i (`'R"B)) /\ xRy))
17	16	19.22i 1040	. . 3 |- (E.x((x e. A /\ xRy) /\ y e. B) -> E.x(x e. (A i^i (`'R"B)) /\ xRy))
18	2	elima2 3409	. . . . 5 |- (y e. (R"A) <-> E.x(x e. A /\ xRy))
19	18	anbi1i 481	. . . 4 |- ((y e. (R"A) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
20		elin 2207	. . . 4 |- (y e. ((R"A) i^i B) <-> (y e. (R"A) /\ y e. B))
21		19.41v 1305	. . . 4 |- (E.x((x e. A /\ xRy) /\ y e. B) <-> (E.x(x e. A /\ xRy) /\ y e. B))
22	19, 20, 21	3bitr4 183	. . 3 |- (y e. ((R"A) i^i B) <-> E.x((x e. A /\ xRy) /\ y e. B))
23	2	elima2 3409	. . 3 |- (y e. (R"(A i^i (`'R"B))) <-> E.x(x e. (A i^i (`'R"B)) /\ xRy))
24	17, 22, 23	3imtr4 219	. 2 |- (y e. ((R"A) i^i B) -> y e. (R"(A i^i (`'R"B))))
25	24	ssriv 2069	1 |- ((R"A) i^i B) (_ (R"(A i^i (`'R"B)))

Syntax hints:   /\ wa 223   e. wcel 958  E.wex 980   i^i cin 2046   (_ wss 2047   class class class wbr 2619  `'ccnv 3169  "cima 3173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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