Theorem f1stres 4093

Description: Mapping of a restriction of the 1st (first member of an ordered pair) function.

Assertion

Ref	Expression
f1stres	|- (1st |` (A X. B)):(A X. B)-->A

Proof of Theorem f1stres

Step	Hyp	Ref	Expression
1		visset 1813	. . . . . . . 8 |- y e. V
2	1	op1sta 3448	. . . . . . 7 |- U.dom {<.y, z>.} = y
3	2	eleq1i 1537	. . . . . 6 |- (U.dom {<.y, z>.} e. A <-> y e. A)
4	3	biimpr 152	. . . . 5 |- (y e. A -> U.dom {<.y, z>.} e. A)
5	4	adantr 389	. . . 4 |- ((y e. A /\ z e. B) -> U.dom {<.y, z>.} e. A)
6	5	rgen2 1723	. . 3 |- A.y e. A A.z e. B U.dom {<.y, z>.} e. A
7		sneq 2417	. . . . . . 7 |- (x = <.y, z>. -> {x} = {<.y, z>.})
8	7	dmeqd 3313	. . . . . 6 |- (x = <.y, z>. -> dom { x} = dom {<.y, z>.})
9	8	unieqd 2512	. . . . 5 |- (x = <.y, z>. -> U.dom { x} = U.dom {<.y, z>.})
10	9	eleq1d 1540	. . . 4 |- (x = <.y, z>. -> (U.dom { x} e. A <-> U.dom {<.y, z>.} e. A))
11	10	ralxp 3218	. . 3 |- (A.x e. (A X. B)U.dom { x} e. A <-> A.y e. A A.z e. B U.dom {<.y, z>.} e. A)
12	6, 11	mpbir 190	. 2 |- A.x e. (A X. B)U.dom { x} e. A
13		df-1st 4079	. . . . 5 |- 1st = {<.x, y>. | y = U.dom { x}}
14		reseq1 3368	. . . . 5 |- (1st = {<.x, y>. | y = U.dom { x}} -> (1st |` (A X. B)) = ({<.x, y>. | y = U.dom { x}} |` (A X. B)))
15	13, 14	ax-mp 7	. . . 4 |- (1st |` (A X. B)) = ({<.x, y>. | y = U.dom { x}} |` (A X. B))
16		resopab 3395	. . . 4 |- ({<.x, y>. | y = U.dom { x}} |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
17	15, 16	eqtr 1495	. . 3 |- (1st |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
18	17	fopab2 3823	. 2 |- (A.x e. (A X. B)U.dom { x} e. A <-> (1st |` (A X. B)):(A X. B)-->A)
19	12, 18	mpbi 189	1 |- (1st |` (A X. B)):(A X. B)-->A

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {csn 2409  <.cop 2411  U.cuni 2503  {copab 2666   X. cxp 3168  dom cdm 3170   |` cres 3172  -->wf 3178  1stc1st 4077

This theorem is referenced by:  1stcof 4101  ruclem17 7526

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-9 965  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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-ral 1649  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-1st 4079

Copyright terms: Public domain