Theorem alephfplem3 4963

Description: Lemma for alephfp 4965.

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 1167	. 2 |- y = y
2		fveq2 3781	. . . 4 |- (v = (/) -> (H` v) = (H` (/)))
3	2	eleq1d 1587	. . 3 |- (v = (/) -> ((H` v) e. ran aleph <-> (H` (/)) e. ran aleph))
4		fveq2 3781	. . . 4 |- (v = w -> (H` v) = (H` w))
5	4	eleq1d 1587	. . 3 |- (v = w -> ((H` v) e. ran aleph <-> (H` w) e. ran aleph))
6		fveq2 3781	. . . 4 |- (v = suc w -> (H` v) = (H` suc w))
7	6	eleq1d 1587	. . 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 4961	. . . 4 |- (H` (/)) e. ran aleph
10	9	a1i 8	. . 3 |- (y = y -> (H` (/)) e. ran aleph)
11	8	alephfplem2 4962	. . . . . 6 |- (w e. om -> (H` suc w) = (aleph` (H` w)))
12	11	eleq1d 1587	. . . . 5 |- (w e. om -> ((H` suc w) e. ran aleph <-> (aleph` (H` w)) e. ran aleph))
13		alephsson 4959	. . . . . . 7 |- ran aleph (_ On
14	13	sseli 2116	. . . . . 6 |- ((H` w) e. ran aleph -> (H` w) e. On)
15		alephfnon 4927	. . . . . . 7 |- aleph Fn On
16		fnfvelrn 3870	. . . . . . 7 |- ((aleph Fn On /\ (H` w) e. On) -> (aleph` (H` w)) e. ran aleph)
17	15, 16	mpan 707	. . . . . 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 217	. . . 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 3215	. 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 997   e. wcel 999  (/)c0 2331  {copab 2721  Oncon0 3005  suc csuc 3007  omcom 3188  ran crn 3228   |` cres 3229   Fn wfn 3234  ` cfv 3239  reccrdg 3989  alephcale 4876

This theorem is referenced by:  alephfplem4 4964  alephfp 4965

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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