Theorem dminss 3448

Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising."

Assertion

Ref	Expression
dminss	|- (dom R i^i A) (_ (`'R"(R"A))

Proof of Theorem dminss

Step	Hyp	Ref	Expression
1		19.8a 1025	. . . . . . 7 |- ((x e. A /\ xRy) -> E.x(x e. A /\ xRy))
2	1	ancoms 436	. . . . . 6 |- ((xRy /\ x e. A) -> E.x(x e. A /\ xRy))
3		visset 1804	. . . . . . 7 |- y e. V
4	3	elima2 3393	. . . . . 6 |- (y e. (R"A) <-> E.x(x e. A /\ xRy))
5	2, 4	sylibr 200	. . . . 5 |- ((xRy /\ x e. A) -> y e. (R"A))
6		pm3.26 319	. . . . . 6 |- ((xRy /\ x e. A) -> xRy)
7		visset 1804	. . . . . . 7 |- x e. V
8	3, 7	brcnv 3288	. . . . . 6 |- (y`'Rx <-> xRy)
9	6, 8	sylibr 200	. . . . 5 |- ((xRy /\ x e. A) -> y`'Rx)
10	5, 9	jca 288	. . . 4 |- ((xRy /\ x e. A) -> (y e. (R"A) /\ y`'Rx))
11	10	19.22i 1036	. . 3 |- (E.y(xRy /\ x e. A) -> E.y(y e. (R"A) /\ y`'Rx))
12	7	eldm 3296	. . . . 5 |- (x e. dom R <-> E.y xRy)
13	12	anbi1i 480	. . . 4 |- ((x e. dom R /\ x e. A) <-> (E.y xRy /\ x e. A))
14		elin 2197	. . . 4 |- (x e. (dom R i^i A) <-> (x e. dom R /\ x e. A))
15		19.41v 1300	. . . 4 |- (E.y(xRy /\ x e. A) <-> (E.y xRy /\ x e. A))
16	13, 14, 15	3bitr4 183	. . 3 |- (x e. (dom R i^i A) <-> E.y(xRy /\ x e. A))
17	7	elima2 3393	. . 3 |- (x e. (`'R"(R"A)) <-> E.y(y e. (R"A) /\ y`'Rx))
18	11, 16, 17	3imtr4 219	. 2 |- (x e. (dom R i^i A) -> x e. (`'R"(R"A)))
19	18	ssriv 2059	1 |- (dom R i^i A) (_ (`'R"(R"A))

Syntax hints:   /\ wa 223   e. wcel 955  E.wex 977   i^i cin 2036   (_ wss 2037   class class class wbr 2609  `'ccnv 3159  dom cdm 3160  "cima 3163

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181

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