Theorem alephfplem4 5049

Description: Lemma for alephfp 5050.

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 3942	. . . 4 |- (H:om-->ran aleph <-> (H Fn om /\ A.z e. om (H` z) e. ran aleph))
2		frfnom 4252	. . . . 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 3688	. . . . . 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 188	. . . 4 |- H Fn om
7	3	alephfplem3 5048	. . . . 5 |- (z e. om -> (H` z) e. ran aleph)
8	7	rgen 1744	. . . 4 |- A.z e. om (H` z) e. ran aleph
9	1, 6, 8	mpbir2an 735	. . 3 |- H:om-->ran aleph
10		ssun2 2246	. . 3 |- ran aleph (_ (om u. ran aleph)
11		fss 3742	. . 3 |- ((H:om-->ran aleph /\ ran aleph (_ (om u. ran aleph)) -> H:om-->(om u. ran aleph))
12	9, 10, 11	mp2an 701	. 2 |- H:om-->(om u. ran aleph)
13		peano1 3237	. . 3 |- (/) e. om
14	3	alephfplem1 5046	. . 3 |- (H` (/)) e. ran aleph
15		fveq2 3835	. . . . 5 |- (z = (/) -> (H` z) = (H` (/)))
16	15	eleq1d 1583	. . . 4 |- (z = (/) -> ((H` z) e. ran aleph <-> (H` (/)) e. ran aleph))
17	16	rcla4ev 1923	. . 3 |- (((/) e. om /\ (H` (/)) e. ran aleph) -> E.z e. om (H` z) e. ran aleph)
18	13, 14, 17	mp2an 701	. 2 |- E.z e. om (H` z) e. ran aleph
19		omex 4772	. . 3 |- om e. V
20		cardinfima 5041	. . 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 701	1 |- U.(H"om) e. ran aleph

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692  Vcvv 1857   u. cun 2097   (_ wss 2099  (/)c0 2332  U.cuni 2569  {copab 2740  omcom 3218  ran crn 3252   |` cres 3253  "cima 3254   Fn wfn 3258  -->wf 3259  ` cfv 3263  reccrdg 4232  alephcale 4960

This theorem is referenced by:  alephfp 5050  alephfp2 5051

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-reg 4736  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-rdg 4233  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-fin 4512  df-card 4962  df-aleph 4963

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