Theorem numth 4756

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 3922	. 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 1468	. 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 59	. . . 4 |- (u = y -> u = y)
5		rneq 3328	. . . . 5 |- (v = f -> ran v = ran f)
6		difeq2 2144	. . . . 5 |- (ran v = ran f -> (A \ ran v) = (A \ ran f))
7		fveq2 3709	. . . . 5 |- ((A \ ran v) = (A \ ran f) -> (h` (A \ ran v)) = (h` (A \ ran f)))
8	5, 6, 7	3syl 20	. . . 4 |- (v = f -> (h` (A \ ran v)) = (h` (A \ ran f)))
9	4, 8	eqeqan12rd 1483	. . 3 |- ((v = f /\ u = y) -> (u = (h` (A \ ran v)) <-> y = (h` (A \ ran f))))
10	9	cbvopabv 2663	. 2 |- {<.v, u>. | u = (h` (A \ ran v))} = {<.f, y>. | y = (h` (A \ ran f))}
11	1, 2, 3, 10	numthlem 4755	1 |- E.x e. On E.f f:x-1-1-onto->A

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  A.wral 1637  E.wrex 1638  Vcvv 1802   \ cdif 2034  U.cuni 2493  {copab 2656  Oncon0 2938  ran crn 3161   |` cres 3162   Fn wfn 3167  -1-1-onto->wf1o 3171  ` cfv 3172

This theorem is referenced by:  numth2 4757  weth 4759

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  ax-ac 4716

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-reu 1643  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-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  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-suc 2944  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188

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