Theorem infxpidmlem9 7511

Description: Lemma for infxpidm 7515. By Zorn's Lemma zorn 4777, 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 2263	. . . . 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 1495	. . . 4 |- H = ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)})
4		df-sn 2408	. . . . . 6 |- {(/)} = {f | f = (/)}
5		p0ex 2765	. . . . . 6 |- {(/)} e. V
6	4, 5	eqeltrr 1542	. . . . 5 |- {f | f = (/)} e. V
7		df-rex 1647	. . . . . . . 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 1809	. . . . . . . . . . . 12 |- t e. V
9	8	elpw 2400	. . . . . . . . . . 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 1049	. . . . . . . 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 1572	. . . . . 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 2740	. . . . . . 7 |- P~A e. V
19	8, 8	xpex 3255	. . . . . . . . 9 |- (t X. t) e. V
20		mapex 4318	. . . . . . . . 9 |- (((t X. t) e. V /\ t e. V) -> {f | f:(t X. t)-->t} e. V)
21	19, 8, 20	mp2an 696	. . . . . . . 8 |- {f | f:(t X. t)-->t} e. V
22		f1of 3680	. . . . . . . . . 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 2116	. . . . . . . 8 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} (_ {f | f:(t X. t)-->t}
25	21, 24	ssexi 2715	. . . . . . 7 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
26	18, 25	abrexex2 3862	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
27	16, 26	eqeltrr 1542	. . . . 5 |- {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)} e. V
28	6, 27	unex 2867	. . . 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 1541	. . 3 |- H e. V
30	29	zorn 4777	. 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 1473	. . 3 |- ran U. z = ran U. z
32		visset 1809	. . 3 |- z e. V
33	1, 31, 32	infxpidmlem8 7510	. 2 |- ((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H)
34	30, 33	mpg 984	1 |- E.g e. H A.h e. H -. g (. h

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  A.wral 1642  E.wrex 1643  Vcvv 1807   u. cun 2041   (_ wss 2043   (. wpss 2044  (/)c0 2276  P~cpw 2397  {csn 2405  U.cuni 2498   class class class wbr 2614  omcom 3126   X. cxp 3163  ran crn 3166  -->wf 3173  -1-1-onto->wf1o 3176   ~<_ cdom 4355

This theorem is referenced by:  infxpidmlem12 7514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-iso 3194  df-en 4357  df-dom 4358

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