Theorem alephfplem4 6047

Description: Lemma for alephfp 6048.

Hypothesis

Ref	Expression
alephfplem.1	|- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)

Assertion

Ref	Expression
alephfplem4	|- U.(H"om) e. ran aleph

Distinct variable group:   x,y

Proof of Theorem alephfplem4

Step	Hyp	Ref	Expression
1		ffnfv 4801	. . . 4 |- (H:om-->ran aleph <-> (H Fn om /\ A.z e. om (H` z) e. ran aleph))
2		frfnom 5159	. . . . 5 |- (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om
3		alephfplem.1	. . . . . 6 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
4	3	fneq1i 4507	. . . . 5 |- (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om)
5	2, 4	mpbir 207	. . . 4 |- H Fn om
6	3	alephfplem3 6046	. . . . 5 |- (z e. om -> (H` z) e. ran aleph)
7	6	rgen 2159	. . . 4 |- A.z e. om (H` z) e. ran aleph
8	1, 5, 7	mpbir2an 800	. . 3 |- H:om-->ran aleph
9		ssun2 2768	. . 3 |- ran aleph C_ (om u. ran aleph)
10		fss 4571	. . 3 |- ((H:om-->ran aleph /\ ran aleph C_ (om u. ran aleph)) -> H:om-->(om u. ran aleph))
11	8, 9, 10	mp2an 761	. 2 |- H:om-->(om u. ran aleph)
12		peano1 3971	. . 3 |- (/) e. om
13	3	alephfplem1 6044	. . 3 |- (H` (/)) e. ran aleph
14		fveq2 4681	. . . . 5 |- (z = (/) -> (H` z) = (H` (/)))
15	14	eleq1d 1963	. . . 4 |- (z = (/) -> ((H` z) e. ran aleph <-> (H` (/)) e. ran aleph))
16	15	rcla4ev 2381	. . 3 |- (((/) e. om /\ (H` (/)) e. ran aleph) -> E.z e. om (H` z) e. ran aleph)
17	12, 13, 16	mp2an 761	. 2 |- E.z e. om (H` z) e. ran aleph
18		omex 5733	. . 3 |- om e. _V
19		cardinfima 6039	. . 3 |- (om e. _V -> ((H:om-->(om u. ran aleph) /\ E.z e. om (H` z) e. ran aleph) -> U.(H"om) e. ran aleph))
20	18, 19	ax-mp 7	. 2 |- ((H:om-->(om u. ran aleph) /\ E.z e. om (H` z) e. ran aleph) -> U.(H"om) e. ran aleph)
21	11, 17, 20	mp2an 761	1 |- U.(H"om) e. ran aleph

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  U.cuni 3177  {copab 3395  omcom 3949  ran crn 3987   |` cres 3988  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  reccrdg 5139  alephcale 5860

This theorem is referenced by:  alephfp 6048  alephfp2 6049

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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