Theorem rdg0 3947

Description: The initial value of the recursive definition generator.

Hypothesis

Ref	Expression
rdg.1	⊢ A ∈ V

Assertion

Ref	Expression
rdg0	⊢ (rec(F, A) ‘∅) = A

Proof of Theorem rdg0

Step	Hyp	Ref	Expression
1		rdglem2 3944	. 2 ⊢ {⟨w, z⟩∣((w = ∅ ⋀ z = A) ⋁ (¬ (w = ∅ ⋁ Lim dom w) ⋀ z = (F ‘(w ‘∪dom w))) ⋁ (Lim dom w ⋀ z = ∪ran w))} = {⟨g, z⟩∣((g = ∅ ⋀ z = A) ⋁ (¬ (g = ∅ ⋁ Lim dom g) ⋀ z = (F ‘(g ‘∪dom g))) ⋁ (Lim dom g ⋀ z = ∪ran g))}
2		rdgfnon 3945	. 2 ⊢ rec(F, A) Fn On
3		rdgval 3946	. 2 ⊢ (g ∈ On → (rec(F, A) ‘g) = ({⟨w, z⟩∣((w = ∅ ⋀ z = A) ⋁ (¬ (w = ∅ ⋁ Lim dom w) ⋀ z = (F ‘(w ‘∪dom w))) ⋁ (Lim dom w ⋀ z = ∪ran w))} ‘(rec(F, A) ↾ g)))
4		rdg.1	. 2 ⊢ A ∈ V
5	1, 2, 3, 4	tz7.44-1 3934	1 ⊢ (rec(F, A) ‘∅) = A

Syntax hints:  ¬ wn 2   ⋁ wo 222   ⋀ wa 223   ⋁ w3o 776   = wceq 958   ∈ wcel 960  Vcvv 1814  ∅c0 2283  ∪cuni 2507  {copab 2671  Lim wlim 2955  dom cdm 3176  ran crn 3177   ‘cfv 3188  reccrdg 3937

This theorem is referenced by:  rdg0t 3950  abianfplem 3967  om0 4162  oe0 4167  oev2 4168  r10 4661  aleph0 4874

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

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-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  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-lim 2959  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-fv 3204  df-rdg 3938

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