Theorem dminss 3454

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 1027	. . . . . . 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 1809	. . . . . . 7 |- y e. V
4	3	elima2 3401	. . . . . 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 1809	. . . . . . 7 |- x e. V
8	3, 7	brcnv 3294	. . . . . 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 1038	. . 3 |- (E.y(xRy /\ x e. A) -> E.y(y e. (R"A) /\ y`'Rx))
12	7	eldm 3302	. . . . 5 |- (x e. dom R <-> E.y xRy)
13	12	anbi1i 481	. . . 4 |- ((x e. dom R /\ x e. A) <-> (E.y xRy /\ x e. A))
14		elin 2203	. . . 4 |- (x e. (dom R i^i A) <-> (x e. dom R /\ x e. A))
15		19.41v 1303	. . . 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 3401	. . 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 2065	1 |- (dom R i^i A) (_ (`'R"(R"A))

Syntax hints:   /\ wa 223   e. wcel 956  E.wex 978   i^i cin 2042   (_ wss 2043   class class class wbr 2614  `'ccnv 3164  dom cdm 3165  "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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