Theorem numth 3599

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 ∈ V

Assertion

Ref	Expression
numth	⊢ ∃x ∈ On ∃f f:x–1-1-onto→A

Distinct variable group(s):   x,f,A

Proof of Theorem numth

Step	Hyp	Ref	Expression
1		numth.1	. 2 ⊢ A ∈ V
2		rdglem1 2975	. 2 ⊢ {g∣∃z ∈ On (g Fn z ∧ ∀w ∈ z (g ‘w) = ({⟨v, u⟩∣u = (h ‘(A ∖ ran v))} ‘(g ↾ w)))} = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = ({⟨v, u⟩∣u = (h ‘(A ∖ ran v))} ‘(f ↾ y)))}
3		cleqid 1102	. 2 ⊢ ∪{g∣∃z ∈ On (g Fn z ∧ ∀w ∈ z (g ‘w) = ({⟨v, u⟩∣u = (h ‘(A ∖ ran v))} ‘(g ↾ w)))} = ∪{g∣∃z ∈ On (g Fn z ∧ ∀w ∈ z (g ‘w) = ({⟨v, u⟩∣u = (h ‘(A ∖ ran v))} ‘(g ↾ w)))}
4		id 9	. . . 4 ⊢ (u = y → u = y)
5		rneq 2555	. . . . 5 ⊢ (v = f → ran v = ran f)
6		difeq2 1583	. . . . 5 ⊢ (ran v = ran f → (A ∖ ran v) = (A ∖ ran f))
7		fveq2 2832	. . . . 5 ⊢ ((A ∖ ran v) = (A ∖ ran f) → (h ‘(A ∖ ran v)) = (h ‘(A ∖ ran f)))
8	5, 6, 7	3syl 21	. . . 4 ⊢ (v = f → (h ‘(A ∖ ran v)) = (h ‘(A ∖ ran f)))
9	4, 8	cleqan12rd 1117	. . 3 ⊢ ((v = f ∧ u = y) → (u = (h ‘(A ∖ ran v)) ↔ y = (h ‘(A ∖ ran f))))
10	9	cbvopabv 2105	. 2 ⊢ {⟨v, u⟩∣u = (h ‘(A ∖ ran v))} = {⟨f, y⟩∣y = (h ‘(A ∖ ran f))}
11	1, 2, 3, 10	numthlem 3598	1 ⊢ ∃x ∈ On ∃f f:x–1-1-onto→A

Syntax hints:   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ∖ cdif 1484  ∪cuni 1919  {copab 2055  Oncon0 2199  ran crn 2411   ↾ cres 2412   Fn wfn 2417  –1-1-onto→wf1o 2421   ‘cfv 2422

This theorem is referenced by:  numth2 3600  weth 3602

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438

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