Theorem numth 4767

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 3932	. 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 1474	. 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 3335	. . . . 5 |- (v = f -> ran v = ran f)
6		difeq2 2151	. . . . 5 |- (ran v = ran f -> (A \ ran v) = (A \ ran f))
7		fveq2 3719	. . . . 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 1489	. . 3 |- ((v = f /\ u = y) -> (u = (h` (A \ ran v)) <-> y = (h` (A \ ran f))))
10	9	cbvopabv 2669	. 2 |- {<.v, u>. | u = (h` (A \ ran v))} = {<.f, y>. | y = (h` (A \ ran f))}
11	1, 2, 3, 10	numthlem 4766	1 |- E.x e. On E.f f:x-1-1-onto->A

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462  A.wral 1643  E.wrex 1644  Vcvv 1808   \ cdif 2041  U.cuni 2499  {copab 2662  Oncon0 2944  ran crn 3167   |` cres 3168   Fn wfn 3173  -1-1-onto->wf1o 3177  ` cfv 3178

This theorem is referenced by:  numth2 4768  weth 4770

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194

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