Theorem f2ndres 4084

Description: Mapping of a restriction of the 2nd (second member of an ordered pair) function.

Assertion

Ref	Expression
f2ndres	|- (2nd |` (A X. B)):(A X. B)-->B

Proof of Theorem f2ndres

Step	Hyp	Ref	Expression
1		visset 1809	. . . . . . . 8 |- y e. V
2		visset 1809	. . . . . . . 8 |- z e. V
3	1, 2	op2nda 3444	. . . . . . 7 |- U.ran {<.y, z>.} = z
4	3	eleq1i 1534	. . . . . 6 |- (U.ran {<.y, z>.} e. B <-> z e. B)
5	4	biimpr 152	. . . . 5 |- (z e. B -> U.ran {<.y, z>.} e. B)
6	5	adantl 388	. . . 4 |- ((y e. A /\ z e. B) -> U.ran {<.y, z>.} e. B)
7	6	rgen2 1720	. . 3 |- A.y e. A A.z e. B U.ran {<.y, z>.} e. B
8		sneq 2413	. . . . . . 7 |- (x = <.y, z>. -> {x} = {<.y, z>.})
9	8	rneqd 3336	. . . . . 6 |- (x = <.y, z>. -> ran { x} = ran {<.y, z>.})
10	9	unieqd 2507	. . . . 5 |- (x = <.y, z>. -> U.ran { x} = U.ran {<.y, z>.})
11	10	eleq1d 1537	. . . 4 |- (x = <.y, z>. -> (U.ran { x} e. B <-> U.ran {<.y, z>.} e. B))
12	11	ralxp 3213	. . 3 |- (A.x e. (A X. B)U.ran { x} e. B <-> A.y e. A A.z e. B U.ran {<.y, z>.} e. B)
13	7, 12	mpbir 190	. 2 |- A.x e. (A X. B)U.ran { x} e. B
14		df-2nd 4070	. . . . 5 |- 2nd = {<.x, y>. | y = U.ran { x}}
15		reseq1 3360	. . . . 5 |- (2nd = {<.x, y>. | y = U.ran { x}} -> (2nd |` (A X. B)) = ({<.x, y>. | y = U.ran { x}} |` (A X. B)))
16	14, 15	ax-mp 7	. . . 4 |- (2nd |` (A X. B)) = ({<.x, y>. | y = U.ran { x}} |` (A X. B))
17		resopab 3387	. . . 4 |- ({<.x, y>. | y = U.ran { x}} |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.ran { x})}
18	16, 17	eqtr 1492	. . 3 |- (2nd |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.ran { x})}
19	18	fopab2 3814	. 2 |- (A.x e. (A X. B)U.ran { x} e. B <-> (2nd |` (A X. B)):(A X. B)-->B)
20	13, 19	mpbi 189	1 |- (2nd |` (A X. B)):(A X. B)-->B

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  {csn 2405  <.cop 2407  U.cuni 2498  {copab 2661   X. cxp 3163  ran crn 3166   |` cres 3167  -->wf 3173  2ndc2nd 4068

This theorem is referenced by:  2ndconst 4087

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-9 963  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-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

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-ral 1646  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-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-2nd 4070

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