Theorem alephfplem4 4899

Description: Lemma for alephfp 4900.

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 3828	. . . 4 |- (H:om-->ran aleph <-> (H Fn om /\ A.z e. om (H` z) e. ran aleph))
2		frfnom 3951	. . . . 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		fneq1 3582	. . . . . 6 |- (H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) -> (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om))
5	3, 4	ax-mp 7	. . . . 5 |- (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om)
6	2, 5	mpbir 190	. . . 4 |- H Fn om
7	3	alephfplem3 4898	. . . . 5 |- (z e. om -> (H` z) e. ran aleph)
8	7	rgen 1698	. . . 4 |- A.z e. om (H` z) e. ran aleph
9	1, 6, 8	mpbir2an 730	. . 3 |- H:om-->ran aleph
10		ssun2 2194	. . 3 |- ran aleph (_ (om u. ran aleph)
11		fss 3635	. . 3 |- ((H:om-->ran aleph /\ ran aleph (_ (om u. ran aleph)) -> H:om-->(om u. ran aleph))
12	9, 10, 11	mp2an 697	. 2 |- H:om-->(om u. ran aleph)
13		peano1 3149	. . 3 |- (/) e. om
14	3	alephfplem1 4896	. . 3 |- (H` (/)) e. ran aleph
15		fveq2 3724	. . . . 5 |- (z = (/) -> (H` z) = (H` (/)))
16	15	eleq1d 1540	. . . 4 |- (z = (/) -> ((H` z) e. ran aleph <-> (H` (/)) e. ran aleph))
17	16	rcla4ev 1877	. . 3 |- (((/) e. om /\ (H` (/)) e. ran aleph) -> E.z e. om (H` z) e. ran aleph)
18	13, 14, 17	mp2an 697	. 2 |- E.z e. om (H` z) e. ran aleph
19		omex 4627	. . 3 |- om e. V
20		cardinfima 4891	. . 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))
21	19, 20	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)
22	12, 18, 21	mp2an 697	1 |- U.(H"om) e. ran aleph

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   u. cun 2045   (_ wss 2047  (/)c0 2280  U.cuni 2503  {copab 2666  omcom 3131  ran crn 3171   |` cres 3172  "cima 3173   Fn wfn 3177  -->wf 3178  ` cfv 3182  reccrdg 3931  alephcale 4814

This theorem is referenced by:  alephfp 4900  alephfp2 4901

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-reg 4593  ax-inf2 4625  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-if 2362  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-lim 2953  df-suc 2954  df-om 3132  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-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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