Theorem infxpidmlem2 7553

Description: Lemma for infxpidm 7564. 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 1481	. . . 4 |- (f = B -> (f = (/) <-> B = (/)))
3		f1oeq1 3684	. . . . . 6 |- (f = B -> (f:(t X. t)-1-1-onto->t <-> B:(t X. t)-1-1-onto->t))
4	3	anbi2d 616	. . . . 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 1279	. . . 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 627	. . 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 1901	. 2 |- (B e. H <-> (B = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ B:(t X. t)-1-1-onto->t)))
9		breq2 2623	. . . . . 6 |- (x = t -> (om ~<_ x <-> om ~<_ t))
10		sseq1 2082	. . . . . 6 |- (x = t -> (x (_ A <-> t (_ A))
11	9, 10	anbi12d 628	. . . . 5 |- (x = t -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ t /\ t (_ A)))
12		xpeq1 3200	. . . . . . . 8 |- (x = t -> (x X. x) = (t X. x))
13		xpeq2 3201	. . . . . . . 8 |- (x = t -> (t X. x) = (t X. t))
14	12, 13	eqtrd 1507	. . . . . . 7 |- (x = t -> (x X. x) = (t X. t))
15		f1oeq2 3685	. . . . . . 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 3686	. . . . . 6 |- (x = t -> (B:(t X. t)-1-1-onto->x <-> B:(t X. t)-1-1-onto->t))
18	16, 17	bitrd 528	. . . . 5 |- (x = t -> (B:(x X. x)-1-1-onto->x <-> B:(t X. t)-1-1-onto->t))
19	11, 18	anbi12d 628	. . . 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 1315	. . 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 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811   (_ wss 2047  (/)c0 2280   class class class wbr 2619  omcom 3131   X. cxp 3168  -1-1-onto->wf1o 3181   ~<_ cdom 4365

This theorem is referenced by:  infxpidmlem3 7554  infxpidmlem4 7555  infxpidmlem7 7558  infxpidmlem8 7559  infxpidmlem12 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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