Theorem f1stres 5034

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 2295	. . . . . . . 8 |- y e. _V
2	1	op1sta 4372	. . . . . . 7 |- U.dom {<.y, z>.} = y
3	2	eleq1i 1960	. . . . . 6 |- (U.dom {<.y, z>.} e. A <-> y e. A)
4	3	biimpri 169	. . . . 5 |- (y e. A -> U.dom {<.y, z>.} e. A)
5	4	adantr 425	. . . 4 |- ((y e. A /\ z e. B) -> U.dom {<.y, z>.} e. A)
6	5	rgen2 2186	. . 3 |- A.y e. A A.z e. B U.dom {<.y, z>.} e. A
7		sneq 3054	. . . . . . 7 |- (x = <.y, z>. -> {x} = {<.y, z>.})
8	7	dmeqd 4159	. . . . . 6 |- (x = <.y, z>. -> dom { x} = dom {<.y, z>.})
9	8	unieqd 3188	. . . . 5 |- (x = <.y, z>. -> U.dom { x} = U.dom {<.y, z>.})
10	9	eleq1d 1963	. . . 4 |- (x = <.y, z>. -> (U.dom { x} e. A <-> U.dom {<.y, z>.} e. A))
11	10	ralxp 4041	. . 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 207	. 2 |- A.x e. (A X. B)U.dom { x} e. A
13		df-1st 5020	. . . . 5 |- 1st = {<.x, y>. | y = U.dom { x}}
14		reseq1 4218	. . . . 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 4252	. . . 4 |- ({<.x, y>. | y = U.dom { x}} |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
17	15, 16	eqtri 1908	. . 3 |- (1st |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
18	17	fopab2 4796	. 2 |- (A.x e. (A X. B)U.dom { x} e. A <-> (1st |` (A X. B)):(A X. B)-->A)
19	12, 18	mpbi 206	1 |- (1st |` (A X. B)):(A X. B)-->A

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {csn 3044  <.cop 3046  U.cuni 3177  {copab 3395   X. cxp 3984  dom cdm 3986   |` cres 3988  -->wf 3994  1stc1st 5018

This theorem is referenced by:  fo1stres 5036  1stcof 5040  ruclem17 8795  tx1cn 10223  algrf 13739  algr0 13740  algrp1lem 13741  algrp1 13742  filnet 15645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-1st 5020

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