Theorem imainss 3455

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 1027	. . . . . . . . . 10 |- ((y e. B /\ y`'Rx) -> E.y(y e. B /\ y`'Rx))
2		visset 1809	. . . . . . . . . . 11 |- y e. V
3		visset 1809	. . . . . . . . . . 11 |- x e. V
4	2, 3	brcnv 3294	. . . . . . . . . 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 2203	. . . . . . 7 |- (x e. (A i^i (`'R"B)) <-> (x e. A /\ x e. (`'R"B)))
12	3	elima2 3401	. . . . . . . 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 1038	. . 3 |- (E.x((x e. A /\ xRy) /\ y e. B) -> E.x(x e. (A i^i (`'R"B)) /\ xRy))
18	2	elima2 3401	. . . . 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 2203	. . . 4 |- (y e. ((R"A) i^i B) <-> (y e. (R"A) /\ y e. B))
21		19.41v 1303	. . . 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 3401	. . 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 2065	1 |- ((R"A) i^i B) (_ (R"(A i^i (`'R"B)))

Syntax hints:   /\ wa 223   e. wcel 956  E.wex 978   i^i cin 2042   (_ wss 2043   class class class wbr 2614  `'ccnv 3164  "cima 3168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186

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