Theorem alephfplem4 4964

Description: Lemma for alephfp 4965.

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 3885	. . . 4 |- (H:om-->ran aleph <-> (H Fn om /\ A.z e. om (H` z) e. ran aleph))
2		frfnom 4009	. . . . 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 3639	. . . . . 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 197	. . . 4 |- H Fn om
7	3	alephfplem3 4963	. . . . 5 |- (z e. om -> (H` z) e. ran aleph)
8	7	rgen 1745	. . . 4 |- A.z e. om (H` z) e. ran aleph
9	1, 6, 8	mpbir2an 742	. . 3 |- H:om-->ran aleph
10		ssun2 2245	. . 3 |- ran aleph (_ (om u. ran aleph)
11		fss 3692	. . 3 |- ((H:om-->ran aleph /\ ran aleph (_ (om u. ran aleph)) -> H:om-->(om u. ran aleph))
12	9, 10, 11	mp2an 709	. 2 |- H:om-->(om u. ran aleph)
13		peano1 3206	. . 3 |- (/) e. om
14	3	alephfplem1 4961	. . 3 |- (H` (/)) e. ran aleph
15		fveq2 3781	. . . . 5 |- (z = (/) -> (H` z) = (H` (/)))
16	15	eleq1d 1587	. . . 4 |- (z = (/) -> ((H` z) e. ran aleph <-> (H` (/)) e. ran aleph))
17	16	rcla4ev 1924	. . 3 |- (((/) e. om /\ (H` (/)) e. ran aleph) -> E.z e. om (H` z) e. ran aleph)
18	13, 14, 17	mp2an 709	. 2 |- E.z e. om (H` z) e. ran aleph
19		omex 4689	. . 3 |- om e. V
20		cardinfima 4956	. . 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 709	1 |- U.(H"om) e. ran aleph

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  A.wral 1692  E.wrex 1693  Vcvv 1858   u. cun 2096   (_ wss 2098  (/)c0 2331  U.cuni 2557  {copab 2721  omcom 3188  ran crn 3228   |` cres 3229  "cima 3230   Fn wfn 3234  -->wf 3235  ` cfv 3239  reccrdg 3989  alephcale 4876

This theorem is referenced by:  alephfp 4965  alephfp2 4966

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-reg 4653  ax-inf2 4687  ax-ac 4806

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-reu 1698  df-rab 1699  df-v 1859  df-sbc 1989  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-pss 2106  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-f1 3252  df-fo 3253  df-f1o 3254  df-fv 3255  df-rdg 3990  df-er 4319  df-en 4429  df-dom 4430  df-sdom 4431  df-fin 4432  df-card 4878  df-aleph 4879

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