Theorem alephfplem3 4821

Description: Lemma for alephfp 4823.

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 1113	. 2 |- y = y
2		fveq2 3663	. . . 4 |- (v = (/) -> (H` v) = (H` (/)))
3	2	eleq1d 1516	. . 3 |- (v = (/) -> ((H` v) e. ran aleph <-> (H` (/)) e. ran aleph))
4		fveq2 3663	. . . 4 |- (v = w -> (H` v) = (H` w))
5	4	eleq1d 1516	. . 3 |- (v = w -> ((H` v) e. ran aleph <-> (H` w) e. ran aleph))
6		fveq2 3663	. . . 4 |- (v = suc w -> (H` v) = (H` suc w))
7	6	eleq1d 1516	. . 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 4819	. . . 4 |- (H` (/)) e. ran aleph
10	9	a1i 8	. . 3 |- (y = y -> (H` (/)) e. ran aleph)
11	8	alephfplem2 4820	. . . . . 6 |- (w e. om -> (H` suc w) = (aleph` (H` w)))
12	11	eleq1d 1516	. . . . 5 |- (w e. om -> ((H` suc w) e. ran aleph <-> (aleph` (H` w)) e. ran aleph))
13		alephsson 4817	. . . . . . 7 |- ran aleph (_ On
14	13	sseli 2036	. . . . . 6 |- ((H` w) e. ran aleph -> (H` w) e. On)
15		alephfnon 4785	. . . . . . 7 |- aleph Fn On
16		fnfvelrn 3752	. . . . . . 7 |- ((aleph Fn On /\ (H` w) e. On) -> (aleph` (H` w)) e. ran aleph)
17	15, 16	mpan 692	. . . . . 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 3121	. 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 1099   e. wcel 1105  (/)c0 2251  {copab 2634  Oncon0 2911  suc csuc 2913  omcom 3094  ran crn 3134   |` cres 3135   Fn wfn 3140  ` cfv 3145  reccrdg 3870  alephcale 4738

This theorem is referenced by:  alephfplem4 4822  alephfp 4823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-reg 4517  ax-inf2 4549  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-rdg 3871  df-er 4199  df-en 4305  df-dom 4306  df-sdom 4307  df-card 4740  df-aleph 4741

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