Theorem numth 4794

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:   x,f,A

Proof of Theorem numth

Step	Hyp	Ref	Expression
1		numth.1	. 2 ⊢ A ∈ V
2		rdglem1 3943	. 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		eqid 1478	. 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 59	. . . 4 ⊢ (u = y → u = y)
5		rneq 3345	. . . . 5 ⊢ (v = f → ran v = ran f)
6		difeq2 2157	. . . . 5 ⊢ (ran v = ran f → (A ∖ ran v) = (A ∖ ran f))
7		fveq2 3730	. . . . 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 1494	. . 3 ⊢ ((v = f ⋀ u = y) → (u = (h ‘(A ∖ ran v)) ↔ y = (h ‘(A ∖ ran f))))
10	9	cbvopabv 2678	. 2 ⊢ {⟨v, u⟩∣u = (h ‘(A ∖ ran v))} = {⟨f, y⟩∣y = (h ‘(A ∖ ran f))}
11	1, 2, 3, 10	numthlem 4793	1 ⊢ ∃x ∈ On ∃f f:x–1-1-onto→A

Syntax hints:   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  {cab 1466  ∀wral 1648  ∃wrex 1649  Vcvv 1814   ∖ cdif 2047  ∪cuni 2507  {copab 2671  Oncon0 2954  ran crn 3177   ↾ cres 3178   Fn wfn 3183  –1-1-onto→wf1o 3187   ‘cfv 3188

This theorem is referenced by:  numth2 4795  weth 4797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204

Copyright terms: Public domain