Theorem infxpidmlem2 7496

Description: Lemma for infxpidm 7507. A necessary and sufficient condition for a set B to belong to H.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem2.2	|- B e. V

Assertion

Ref	Expression
infxpidmlem2	|- (B e. H <-> (B = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x)))

Distinct variable groups:   x,f,t,A   x,B,f,t   x,H

Proof of Theorem infxpidmlem2

Step	Hyp	Ref	Expression
1		infxpidmlem2.2	. . 3 |- B e. V
2		eqeq1 1473	. . . 4 |- (f = B -> (f = (/) <-> B = (/)))
3		f1oeq1 3669	. . . . . 6 |- (f = B -> (f:(t X. t)-1-1-onto->t <-> B:(t X. t)-1-1-onto->t))
4	3	anbi2d 614	. . . . 5 |- (f = B -> (((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t) <-> ((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
5	4	exbidv 1274	. . . 4 |- (f = B -> (E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t) <-> E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
6	2, 5	orbi12d 625	. . 3 |- (f = B -> ((f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)) <-> (B = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t))))
7		infxpidmlem.1	. . 3 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
8	1, 6, 7	elab2 1892	. 2 |- (B e. H <-> (B = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
9		breq2 2613	. . . . . 6 |- (x = t -> (om ~<_ x <-> om ~<_ t))
10		sseq1 2072	. . . . . 6 |- (x = t -> (x (_ A <-> t (_ A))
11	9, 10	anbi12d 626	. . . . 5 |- (x = t -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ t /\ t (_ A)))
12		xpeq1 3190	. . . . . . . 8 |- (x = t -> (x X. x) = (t X. x))
13		xpeq2 3191	. . . . . . . 8 |- (x = t -> (t X. x) = (t X. t))
14	12, 13	eqtrd 1499	. . . . . . 7 |- (x = t -> (x X. x) = (t X. t))
15		f1oeq2 3670	. . . . . . 7 |- ((x X. x) = (t X. t) -> (B:(x X. x)-1-1-onto->x <-> B:(t X. t)-1-1-onto->x))
16	14, 15	syl 10	. . . . . 6 |- (x = t -> (B:(x X. x)-1-1-onto->x <-> B:(t X. t)-1-1-onto->x))
17		f1oeq3 3671	. . . . . 6 |- (x = t -> (B:(t X. t)-1-1-onto->x <-> B:(t X. t)-1-1-onto->t))
18	16, 17	bitrd 526	. . . . 5 |- (x = t -> (B:(x X. x)-1-1-onto->x <-> B:(t X. t)-1-1-onto->t))
19	11, 18	anbi12d 626	. . . 4 |- (x = t -> (((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x) <-> ((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
20	19	cbvexv 1310	. . 3 |- (E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x) <-> E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t))
21	20	orbi2i 255	. 2 |- ((B = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x)) <-> (B = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
22	8, 21	bitr4 176	1 |- (B e. H <-> (B = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x)))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  Vcvv 1802   (_ wss 2037  (/)c0 2270   class class class wbr 2609  omcom 3121   X. cxp 3158  -1-1-onto->wf1o 3171   ~<_ cdom 4349

This theorem is referenced by:  infxpidmlem3 7497  infxpidmlem4 7498  infxpidmlem7 7501  infxpidmlem8 7502  infxpidmlem12 7506

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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