Theorem alephfplem1 4896

Description: Lemma for alephfp 4900.

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 3725	. . 3 |- (H` (/)) = ((rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)` (/))
3		fvex 3732	. . . 4 |- (aleph` (/)) e. V
4		fr0t 3952	. . . 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 1495	. 2 |- (H` (/)) = (aleph` (/))
7		alephfnon 4862	. . 3 |- aleph Fn On
8		0elon 3022	. . 3 |- (/) e. On
9		fnfvelrn 3813	. . 3 |- ((aleph Fn On /\ (/) e. On) -> (aleph` (/)) e. ran aleph)
10	7, 8, 9	mp2an 697	. 2 |- (aleph` (/)) e. ran aleph
11	6, 10	eqeltr 1544	1 |- (H` (/)) e. ran aleph

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280  {copab 2666  Oncon0 2948  omcom 3131  ran crn 3171   |` cres 3172   Fn wfn 3177  ` cfv 3182  reccrdg 3931  alephcale 4814

This theorem is referenced by:  alephfplem3 4898  alephfplem4 4899

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

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-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  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-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-fv 3198  df-rdg 3932  df-aleph 4817

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