Theorem fo1stres 5036

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

Assertion

Ref	Expression
fo1stres	|- (B =/= (/) -> (1st |` (A X. B)):(A X. B)-onto->A)

Proof of Theorem fo1stres

Step	Hyp	Ref	Expression
1		n0 2884	. . . . . . 7 |- (B =/= (/) <-> E.y y e. B)
2		visset 2295	. . . . . . . . . . 11 |- y e. _V
3	2	opelxp 4036	. . . . . . . . . 10 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
4		fvres 4691	. . . . . . . . . . . 12 |- (<.x, y>. e. (A X. B) -> ((1st |` (A X. B))` <.x, y>.) = (1st` <.x, y>.))
5		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
6	5	op1st 5026	. . . . . . . . . . . 12 |- (1st` <.x, y>.) = x
7	4, 6	syl6req 1945	. . . . . . . . . . 11 |- (<.x, y>. e. (A X. B) -> x = ((1st |` (A X. B))` <.x, y>.))
8		f1stres 5034	. . . . . . . . . . . . 13 |- (1st |` (A X. B)):(A X. B)-->A
9		ffn 4562	. . . . . . . . . . . . 13 |- ((1st |` (A X. B)):(A X. B)-->A -> (1st |` (A X. B)) Fn (A X. B))
10	8, 9	ax-mp 7	. . . . . . . . . . . 12 |- (1st |` (A X. B)) Fn (A X. B)
11		fnfvelrn 4786	. . . . . . . . . . . 12 |- (((1st |` (A X. B)) Fn (A X. B) /\ <.x, y>. e. (A X. B)) -> ((1st |` (A X. B))` <.x, y>.) e. ran (1st |` (A X. B)))
12	10, 11	mpan 759	. . . . . . . . . . 11 |- (<.x, y>. e. (A X. B) -> ((1st |` (A X. B))` <.x, y>.) e. ran (1st |` (A X. B)))
13	7, 12	eqeltrd 1971	. . . . . . . . . 10 |- (<.x, y>. e. (A X. B) -> x e. ran (1st |` (A X. B)))
14	3, 13	sylbir 218	. . . . . . . . 9 |- ((x e. A /\ y e. B) -> x e. ran (1st |` (A X. B)))
15	14	expcom 403	. . . . . . . 8 |- (y e. B -> (x e. A -> x e. ran (1st |` (A X. B))))
16	15	19.23aiv 1674	. . . . . . 7 |- (E.y y e. B -> (x e. A -> x e. ran (1st |` (A X. B))))
17	1, 16	sylbi 216	. . . . . 6 |- (B =/= (/) -> (x e. A -> x e. ran (1st |` (A X. B))))
18	17	ssrdv 2622	. . . . 5 |- (B =/= (/) -> A C_ ran (1st |` (A X. B)))
19		frn 4569	. . . . . 6 |- ((1st |` (A X. B)):(A X. B)-->A -> ran (1st |` (A X. B)) C_ A)
20	8, 19	ax-mp 7	. . . . 5 |- ran (1st |` (A X. B)) C_ A
21	18, 20	jctil 316	. . . 4 |- (B =/= (/) -> (ran (1st |` (A X. B)) C_ A /\ A C_ ran (1st |` (A X. B))))
22		eqss 2631	. . . 4 |- (ran (1st |` (A X. B)) = A <-> (ran (1st |` (A X. B)) C_ A /\ A C_ ran (1st |` (A X. B))))
23	21, 22	sylibr 217	. . 3 |- (B =/= (/) -> ran (1st |` (A X. B)) = A)
24	23, 8	jctil 316	. 2 |- (B =/= (/) -> ((1st |` (A X. B)):(A X. B)-->A /\ ran (1st |` (A X. B)) = A))
25		dffo2 4621	. 2 |- ((1st |` (A X. B)):(A X. B)-onto->A <-> ((1st |` (A X. B)):(A X. B)-->A /\ ran (1st |` (A X. B)) = A))
26	24, 25	sylibr 217	1 |- (B =/= (/) -> (1st |` (A X. B)):(A X. B)-onto->A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017   C_ wss 2593  (/)c0 2875  <.cop 3046   X. cxp 3984  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  1stc1st 5018

This theorem is referenced by:  1stconst 5070

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-fo 4012  df-fv 4014  df-1st 5020

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