Theorem f1stres 4099

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 1816	. . . . . . . 8 |- y e. V
2	1	op1sta 3454	. . . . . . 7 |- U.dom {<.y, z>.} = y
3	2	eleq1i 1540	. . . . . 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 391	. . . 4 |- ((y e. A /\ z e. B) -> U.dom {<.y, z>.} e. A)
6	5	rgen2 1726	. . 3 |- A.y e. A A.z e. B U.dom {<.y, z>.} e. A
7		sneq 2421	. . . . . . 7 |- (x = <.y, z>. -> {x} = {<.y, z>.})
8	7	dmeqd 3319	. . . . . 6 |- (x = <.y, z>. -> dom { x} = dom {<.y, z>.})
9	8	unieqd 2516	. . . . 5 |- (x = <.y, z>. -> U.dom { x} = U.dom {<.y, z>.})
10	9	eleq1d 1543	. . . 4 |- (x = <.y, z>. -> (U.dom { x} e. A <-> U.dom {<.y, z>.} e. A))
11	10	ralxp 3224	. . 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 4085	. . . . 5 |- 1st = {<.x, y>. | y = U.dom { x}}
14		reseq1 3374	. . . . 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 3401	. . . 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 1498	. . 3 |- (1st |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
18	17	fopab2 3829	. 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 958   e. wcel 960  A.wral 1648  {csn 2413  <.cop 2415  U.cuni 2507  {copab 2671   X. cxp 3174  dom cdm 3176   |` cres 3178  -->wf 3184  1stc1st 4083

This theorem is referenced by:  1stcof 4107  ruclem17 7527

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-1st 4085

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