Theorem alephfplem1 4868

Description: Lemma for alephfp 4872.

Hypothesis

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

Assertion

Ref	Expression
alephfplem1	|- (H` (/)) e. ran aleph

Distinct variable group:   x,y

Proof of Theorem alephfplem1

Step	Hyp	Ref	Expression
1		alephfplem.1	. . . 4 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
2	1	fveq1i 3710	. . 3 |- (H` (/)) = ((rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)` (/))
3		fvex 3717	. . . 4 |- (aleph` (/)) e. V
4		fr0t 3937	. . . 4 |- ((aleph` (/)) e. V -> ((rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)` (/)) = (aleph` (/)))
5	3, 4	ax-mp 7	. . 3 |- ((rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)` (/)) = (aleph` (/))
6	2, 5	eqtr 1487	. 2 |- (H` (/)) = (aleph` (/))
7		alephfnon 4834	. . 3 |- aleph Fn On
8		0elon 3012	. . 3 |- (/) e. On
9		fnfvelrn 3798	. . 3 |- ((aleph Fn On /\ (/) e. On) -> (aleph` (/)) e. ran aleph)
10	7, 8, 9	mp2an 695	. 2 |- (aleph` (/)) e. ran aleph
11	6, 10	eqeltr 1536	1 |- (H` (/)) e. ran aleph

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  {copab 2656  Oncon0 2938  omcom 3121  ran crn 3161   |` cres 3162   Fn wfn 3167  ` cfv 3172  reccrdg 3916  alephcale 4786

This theorem is referenced by:  alephfplem3 4870  alephfplem4 4871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-aleph 4789

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