Theorem infxpidmlem9 7560

Description: Lemma for infxpidm 7564. By Zorn's Lemma zorn 4797, the collection H (which we show here to be a set) has a maximal element.

Hypotheses

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

Assertion

Ref	Expression
infxpidmlem9	|- E.g e. H A.h e. H -. g (. h

Distinct variable groups:   f,g,h,t,A   g,H,h

Proof of Theorem infxpidmlem9

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . . . 5 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
2		unab 2267	. . . . 5 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3	1, 2	eqtr4 1498	. . . 4 |- H = ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)})
4		df-sn 2412	. . . . . 6 |- {(/)} = {f | f = (/)}
5		p0ex 2770	. . . . . 6 |- {(/)} e. V
6	4, 5	eqeltrr 1545	. . . . 5 |- {f | f = (/)} e. V
7		df-rex 1650	. . . . . . . 8 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
8		visset 1813	. . . . . . . . . . . 12 |- t e. V
9	8	elpw 2404	. . . . . . . . . . 11 |- (t e. P~A <-> t (_ A)
10	9	anbi1i 481	. . . . . . . . . 10 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> (t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
11		ancom 435	. . . . . . . . . 10 |- ((t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A))
12		an23 485	. . . . . . . . . 10 |- (((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
13	10, 11, 12	3bitr 177	. . . . . . . . 9 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
14	13	exbii 1051	. . . . . . . 8 |- (E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
15	7, 14	bitr 173	. . . . . . 7 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
16	15	abbii 1575	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} = {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}
17		infxpidmlem.2	. . . . . . . 8 |- A e. V
18	17	pwex 2745	. . . . . . 7 |- P~A e. V
19	8, 8	xpex 3260	. . . . . . . . 9 |- (t X. t) e. V
20		mapex 4328	. . . . . . . . 9 |- (((t X. t) e. V /\ t e. V) -> {f | f:(t X. t)-->t} e. V)
21	19, 8, 20	mp2an 697	. . . . . . . 8 |- {f | f:(t X. t)-->t} e. V
22		f1of 3689	. . . . . . . . . 10 |- (f:(t X. t)-1-1-onto->t -> f:(t X. t)-->t)
23	22	adantl 388	. . . . . . . . 9 |- ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) -> f:(t X. t)-->t)
24	23	ss2abi 2120	. . . . . . . 8 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} (_ {f | f:(t X. t)-->t}
25	21, 24	ssexi 2720	. . . . . . 7 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
26	18, 25	abrexex2 3871	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
27	16, 26	eqeltrr 1545	. . . . 5 |- {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)} e. V
28	6, 27	unex 2872	. . . 4 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) e. V
29	3, 28	eqeltr 1544	. . 3 |- H e. V
30	29	zorn 4797	. 2 |- (A.z((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H) -> E.g e. H A.h e. H -. g (. h)
31		eqid 1475	. . 3 |- ran U. z = ran U. z
32		visset 1813	. . 3 |- z e. V
33	1, 31, 32	infxpidmlem8 7559	. 2 |- ((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H)
34	30, 33	mpg 986	1 |- E.g e. H A.h e. H -. g (. h

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  A.wral 1645  E.wrex 1646  Vcvv 1811   u. cun 2045   (_ wss 2047   (. wpss 2048  (/)c0 2280  P~cpw 2401  {csn 2409  U.cuni 2503   class class class wbr 2619  omcom 3131   X. cxp 3168  ran crn 3171  -->wf 3178  -1-1-onto->wf1o 3181   ~<_ cdom 4365

This theorem is referenced by:  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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  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-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199  df-en 4368  df-dom 4369

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