Theorem numth 5946

Description: Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84.

Hypothesis

Ref	Expression
numth.1	|- A e. _V

Assertion

Ref	Expression
numth	|- E.x e. On E.f f:x-1-1-onto->A

Distinct variable group:   x,f,A

Proof of Theorem numth

Step	Hyp	Ref	Expression
1		numth.1	. 2 |- A e. _V
2		rdglem1 5145	. 2 |- {g | E.z e. On (g Fn z /\ A.w e. z (g` w) = ({<.v, u>. | u = (h` (A \ ran v))}` (g |` w)))} = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.v, u>. | u = (h` (A \ ran v))}` (f |` y)))}
3		eqid 1884	. 2 |- U.{g | E.z e. On (g Fn z /\ A.w e. z (g` w) = ({<.v, u>. | u = (h` (A \ ran v))}` (g |` w)))} = U.{g | E.z e. On (g Fn z /\ A.w e. z (g` w) = ({<.v, u>. | u = (h` (A \ ran v))}` (g |` w)))}
4		id 73	. . . 4 |- (u = y -> u = y)
5		rneq 4186	. . . . 5 |- (v = f -> ran v = ran f)
6		difeq2 2719	. . . . 5 |- (ran v = ran f -> (A \ ran v) = (A \ ran f))
7		fveq2 4681	. . . . 5 |- ((A \ ran v) = (A \ ran f) -> (h` (A \ ran v)) = (h` (A \ ran f)))
8	5, 6, 7	3syl 24	. . . 4 |- (v = f -> (h` (A \ ran v)) = (h` (A \ ran f)))
9	4, 8	eqeqan12rd 1903	. . 3 |- ((v = f /\ u = y) -> (u = (h` (A \ ran v)) <-> y = (h` (A \ ran f))))
10	9	cbvopabv 3404	. 2 |- {<.v, u>. | u = (h` (A \ ran v))} = {<.f, y>. | y = (h` (A \ ran f))}
11	1, 2, 3, 10	numthlem 5945	1 |- E.x e. On E.f f:x-1-1-onto->A

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590  U.cuni 3177  {copab 3395  Oncon0 3657  ran crn 3987   |` cres 3988   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998

This theorem is referenced by:  numth2 5947  weth 5949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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