Theorem alephfplem3 4898

Description: Lemma for alephfp 4900.

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 1126	. 2 |- y = y
2		fveq2 3724	. . . 4 |- (v = (/) -> (H` v) = (H` (/)))
3	2	eleq1d 1540	. . 3 |- (v = (/) -> ((H` v) e. ran aleph <-> (H` (/)) e. ran aleph))
4		fveq2 3724	. . . 4 |- (v = w -> (H` v) = (H` w))
5	4	eleq1d 1540	. . 3 |- (v = w -> ((H` v) e. ran aleph <-> (H` w) e. ran aleph))
6		fveq2 3724	. . . 4 |- (v = suc w -> (H` v) = (H` suc w))
7	6	eleq1d 1540	. . 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 4896	. . . 4 |- (H` (/)) e. ran aleph
10	9	a1i 8	. . 3 |- (y = y -> (H` (/)) e. ran aleph)
11	8	alephfplem2 4897	. . . . . 6 |- (w e. om -> (H` suc w) = (aleph` (H` w)))
12	11	eleq1d 1540	. . . . 5 |- (w e. om -> ((H` suc w) e. ran aleph <-> (aleph` (H` w)) e. ran aleph))
13		alephsson 4894	. . . . . . 7 |- ran aleph (_ On
14	13	sseli 2065	. . . . . 6 |- ((H` w) e. ran aleph -> (H` w) e. On)
15		alephfnon 4862	. . . . . . 7 |- aleph Fn On
16		fnfvelrn 3813	. . . . . . 7 |- ((aleph Fn On /\ (H` w) e. On) -> (aleph` (H` w)) e. ran aleph)
17	15, 16	mpan 695	. . . . . 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 210	. . . 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 3158	. 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 956   e. wcel 958  (/)c0 2280  {copab 2666  Oncon0 2948  suc csuc 2950  omcom 3131  ran crn 3171   |` cres 3172   Fn wfn 3177  ` cfv 3182  reccrdg 3931  alephcale 4814

This theorem is referenced by:  alephfplem4 4899  alephfp 4900

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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