Theorem alephfplem3 5048

Description: Lemma for alephfp 5050.

Hypothesis

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

Assertion

Ref	Expression
alephfplem3	|- (v e. om -> (H` v) e. ran aleph)

Distinct variable groups:   x,y,v   v,H

Proof of Theorem alephfplem3

Step	Hyp	Ref	Expression
1		equid 1162	. 2 |- y = y
2		fveq2 3835	. . . 4 |- (v = (/) -> (H` v) = (H` (/)))
3	2	eleq1d 1583	. . 3 |- (v = (/) -> ((H` v) e. ran aleph <-> (H` (/)) e. ran aleph))
4		fveq2 3835	. . . 4 |- (v = w -> (H` v) = (H` w))
5	4	eleq1d 1583	. . 3 |- (v = w -> ((H` v) e. ran aleph <-> (H` w) e. ran aleph))
6		fveq2 3835	. . . 4 |- (v = suc w -> (H` v) = (H` suc w))
7	6	eleq1d 1583	. . 3 |- (v = suc w -> ((H` v) e. ran aleph <-> (H` suc w) e. ran aleph))
8		alephfplem.1	. . . . 5 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
9	8	alephfplem1 5046	. . . 4 |- (H` (/)) e. ran aleph
10	9	a1i 8	. . 3 |- (y = y -> (H` (/)) e. ran aleph)
11	8	alephfplem2 5047	. . . . . 6 |- (w e. om -> (H` suc w) = (aleph` (H` w)))
12	11	eleq1d 1583	. . . . 5 |- (w e. om -> ((H` suc w) e. ran aleph <-> (aleph` (H` w)) e. ran aleph))
13		alephsson 5044	. . . . . . 7 |- ran aleph (_ On
14	13	sseli 2117	. . . . . 6 |- ((H` w) e. ran aleph -> (H` w) e. On)
15		alephfnon 5012	. . . . . . 7 |- aleph Fn On
16		fnfvelrn 3927	. . . . . . 7 |- ((aleph Fn On /\ (H` w) e. On) -> (aleph` (H` w)) e. ran aleph)
17	15, 16	mpan 699	. . . . . 6 |- ((H` w) e. On -> (aleph` (H` w)) e. ran aleph)
18	14, 17	syl 10	. . . . 5 |- ((H` w) e. ran aleph -> (aleph` (H` w)) e. ran aleph)
19	12, 18	syl5bir 208	. . . 4 |- (w e. om -> ((H` w) e. ran aleph -> (H` suc w) e. ran aleph))
20	19	a1d 12	. . 3 |- (w e. om -> (y = y -> ((H` w) e. ran aleph -> (H` suc w) e. ran aleph)))
21	3, 5, 7, 10, 20	finds2 3246	. 2 |- (v e. om -> (y = y -> (H` v) e. ran aleph))
22	1, 21	mpi 44	1 |- (v e. om -> (H` v) e. ran aleph)

Syntax hints:   -> wi 3   = wceq 992   e. wcel 994  (/)c0 2332  {copab 2740  Oncon0 2975  suc csuc 2977  omcom 3218  ran crn 3252   |` cres 3253   Fn wfn 3258  ` cfv 3263  reccrdg 4232  alephcale 4960

This theorem is referenced by:  alephfplem4 5049  alephfp 5050

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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