Theorem imfstnrelc 14396

Description: The image under 1st of a class with no pairs inside.

Assertion

Ref	Expression
imfstnrelc	|- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> (1st"A) = {(/)})

Proof of Theorem imfstnrelc

Step	Hyp	Ref	Expression
1		fo1st 5032	. . . . . 6 |- 1st:_V-onto->_V
2		fofun 4618	. . . . . 6 |- (1st:_V-onto->_V -> Fun 1st)
3	1, 2	ax-mp 7	. . . . 5 |- Fun 1st
4		ssv 2636	. . . . . 6 |- A C_ _V
5		fofn 4619	. . . . . . . 8 |- (1st:_V-onto->_V -> 1st Fn _V)
6	1, 5	ax-mp 7	. . . . . . 7 |- 1st Fn _V
7		fndm 4512	. . . . . . 7 |- (1st Fn _V -> dom 1st = _V)
8	6, 7	ax-mp 7	. . . . . 6 |- dom 1st = _V
9	4, 8	sseqtr4i 2650	. . . . 5 |- A C_ dom 1st
10	3, 9	pm3.2i 307	. . . 4 |- (Fun 1st /\ A C_ dom 1st)
11	10	a1i 8	. . 3 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> (Fun 1st /\ A C_ dom 1st))
12		dfimafn2 4721	. . 3 |- ((Fun 1st /\ A C_ dom 1st) -> (1st"A) = U_x e. A {(1st` x)})
13	11, 12	syl 12	. 2 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> (1st"A) = U_x e. A {(1st` x)})
14		disj 2914	. . . . . . . . . . . 12 |- ((A i^i (_V X. _V)) = (/) <-> A.x e. A -. x e. (_V X. _V))
15	14	biimpi 168	. . . . . . . . . . 11 |- ((A i^i (_V X. _V)) = (/) -> A.x e. A -. x e. (_V X. _V))
16		df-ral 2109	. . . . . . . . . . 11 |- (A.x e. A -. x e. (_V X. _V) <-> A.x(x e. A -> -. x e. (_V X. _V)))
17	15, 16	sylib 215	. . . . . . . . . 10 |- ((A i^i (_V X. _V)) = (/) -> A.x(x e. A -> -. x e. (_V X. _V)))
18	17	19.21bi 1408	. . . . . . . . 9 |- ((A i^i (_V X. _V)) = (/) -> (x e. A -> -. x e. (_V X. _V)))
19	18	adantr 425	. . . . . . . 8 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> (x e. A -> -. x e. (_V X. _V)))
20	19	imp 377	. . . . . . 7 |- ((((A i^i (_V X. _V)) = (/) /\ A =/= (/)) /\ x e. A) -> -. x e. (_V X. _V))
21		dmsnn0 4362	. . . . . . . . . 10 |- (x e. (_V X. _V) <-> dom { x} =/= (/))
22	21	bicomi 189	. . . . . . . . 9 |- (dom { x} =/= (/) <-> x e. (_V X. _V))
23	22	necon1bbii 2060	. . . . . . . 8 |- (-. x e. (_V X. _V) <-> dom { x} = (/))
24		0ss 2900	. . . . . . . . 9 |- (/) C_ {(/)}
25		sseq1 2637	. . . . . . . . 9 |- (dom { x} = (/) -> (dom { x} C_ {(/)} <-> (/) C_ {(/)}))
26	24, 25	mpbiri 211	. . . . . . . 8 |- (dom { x} = (/) -> dom { x} C_ {(/)})
27	23, 26	sylbi 216	. . . . . . 7 |- (-. x e. (_V X. _V) -> dom { x} C_ {(/)})
28	20, 27	syl 12	. . . . . 6 |- ((((A i^i (_V X. _V)) = (/) /\ A =/= (/)) /\ x e. A) -> dom { x} C_ {(/)})
29		uni0b 3203	. . . . . 6 |- (U.dom { x} = (/) <-> dom { x} C_ {(/)})
30	28, 29	sylibr 217	. . . . 5 |- ((((A i^i (_V X. _V)) = (/) /\ A =/= (/)) /\ x e. A) -> U.dom { x} = (/))
31		1stval 5022	. . . . 5 |- (1st` x) = U.dom { x}
32	30, 31	syl5eq 1940	. . . 4 |- ((((A i^i (_V X. _V)) = (/) /\ A =/= (/)) /\ x e. A) -> (1st` x) = (/))
33	32	sneqd 3056	. . 3 |- ((((A i^i (_V X. _V)) = (/) /\ A =/= (/)) /\ x e. A) -> {(1st` x)} = {(/)})
34	33	iuneq2dv 3279	. 2 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> U_x e. A {(1st` x)} = U_x e. A {(/)})
35		iunconst 3262	. . 3 |- (A =/= (/) -> U_x e. A {(/)} = {(/)})
36	35	adantl 424	. 2 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> U_x e. A {(/)} = {(/)})
37	13, 34, 36	3eqtrd 1929	1 |- (((A i^i (_V X. _V)) = (/) /\ A =/= (/)) -> (1st"A) = {(/)})

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  U_ciun 3255   X. cxp 3984  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996  ` cfv 3998  1stc1st 5018

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-rab 2112  df-v 2294  df-sbc 2454  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-iun 3257  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-fo 4012  df-fv 4014  df-1st 5020

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