Theorem f1stres 4086

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 1810	. . . . . . . 8 |- y e. V
2	1	op1sta 3444	. . . . . . 7 |- U.dom {<.y, z>.} = y
3	2	eleq1i 1535	. . . . . 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 1721	. . 3 |- A.y e. A A.z e. B U.dom {<.y, z>.} e. A
7		sneq 2414	. . . . . . 7 |- (x = <.y, z>. -> {x} = {<.y, z>.})
8	7	dmeqd 3309	. . . . . 6 |- (x = <.y, z>. -> dom { x} = dom {<.y, z>.})
9	8	unieqd 2508	. . . . 5 |- (x = <.y, z>. -> U.dom { x} = U.dom {<.y, z>.})
10	9	eleq1d 1538	. . . 4 |- (x = <.y, z>. -> (U.dom { x} e. A <-> U.dom {<.y, z>.} e. A))
11	10	ralxp 3214	. . 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 4072	. . . . 5 |- 1st = {<.x, y>. | y = U.dom { x}}
14		reseq1 3364	. . . . 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 3391	. . . 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 1493	. . 3 |- (1st |` (A X. B)) = {<.x, y>. | (x e. (A X. B) /\ y = U.dom { x})}
18	17	fopab2 3818	. 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 955   e. wcel 957  A.wral 1643  {csn 2406  <.cop 2408  U.cuni 2499  {copab 2662   X. cxp 3164  dom cdm 3166   |` cres 3168  -->wf 3174  1stc1st 4070

This theorem is referenced by:  1stcof 4094  ruclem17 7486

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-1st 4072

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