Theorem infxpidmlem3 7554

Description: Lemma for infxpidm 7564. A 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
infxpidmlem3.3	|- D e. V

Assertion

Ref	Expression
infxpidmlem3	|- (((om ~<_ D /\ D (_ A) /\ B:(D X. D)-1-1-onto->D) -> B e. H)

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

Proof of Theorem infxpidmlem3

Step	Hyp	Ref	Expression
1		infxpidmlem3.3	. . 3 |- D e. V
2		breq2 2623	. . . . 5 |- (x = D -> (om ~<_ x <-> om ~<_ D))
3		sseq1 2082	. . . . 5 |- (x = D -> (x (_ A <-> D (_ A))
4	2, 3	anbi12d 628	. . . 4 |- (x = D -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ D /\ D (_ A)))
5		xpeq1 3200	. . . . . . 7 |- (x = D -> (x X. x) = (D X. x))
6		xpeq2 3201	. . . . . . 7 |- (x = D -> (D X. x) = (D X. D))
7	5, 6	eqtrd 1507	. . . . . 6 |- (x = D -> (x X. x) = (D X. D))
8		f1oeq2 3685	. . . . . 6 |- ((x X. x) = (D X. D) -> (B:(x X. x)-1-1-onto->x <-> B:(D X. D)-1-1-onto->x))
9	7, 8	syl 10	. . . . 5 |- (x = D -> (B:(x X. x)-1-1-onto->x <-> B:(D X. D)-1-1-onto->x))
10		f1oeq3 3686	. . . . 5 |- (x = D -> (B:(D X. D)-1-1-onto->x <-> B:(D X. D)-1-1-onto->D))
11	9, 10	bitrd 528	. . . 4 |- (x = D -> (B:(x X. x)-1-1-onto->x <-> B:(D X. D)-1-1-onto->D))
12	4, 11	anbi12d 628	. . 3 |- (x = D -> (((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x) <-> ((om ~<_ D /\ D (_ A) /\ B:(D X. D)-1-1-onto->D)))
13	1, 12	cla4ev 1869	. 2 |- (((om ~<_ D /\ D (_ A) /\ B:(D X. D)-1-1-onto->D) -> E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x))
14		olc 268	. . 3 |- (E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x) -> (B = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x)))
15		infxpidmlem.1	. . . 4 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
16		infxpidmlem2.2	. . . 4 |- B e. V
17	15, 16	infxpidmlem2 7553	. . 3 |- (B e. H <-> (B = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x)))
18	14, 17	sylibr 200	. 2 |- (E.x((om ~<_ x /\ x (_ A) /\ B:(x X. x)-1-1-onto->x) -> B e. H)
19	13, 18	syl 10	1 |- (((om ~<_ D /\ D (_ A) /\ B:(D X. D)-1-1-onto->D) -> B e. H)

Syntax hints:   -> wi 3   <-> 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:  infxpidmlem8 7559  infxpidmlem10 7561  infxpidmlem11 7562

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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