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Theorem bj-rdg0gALT 36255
Description: Alternate proof of rdg0g 8429. More direct since it bypasses tz7.44-1 8408 and rdg0 8423 (and vtoclg 3541, vtoclga 3565). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-rdg0gALT (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Proof of Theorem bj-rdg0gALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rdgdmlim 8419 . . . . 5 Lim dom rec(𝐹, 𝐴)
2 limomss 7862 . . . . 5 (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴))
31, 2ax-mp 5 . . . 4 ω ⊆ dom rec(𝐹, 𝐴)
4 peano1 7881 . . . 4 ∅ ∈ ω
53, 4sselii 3978 . . 3 ∅ ∈ dom rec(𝐹, 𝐴)
6 rdgvalg 8421 . . 3 (∅ ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘∅) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ ∅)))
75, 6ax-mp 5 . 2 (rec(𝐹, 𝐴)‘∅) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ ∅))
8 res0 5984 . . . 4 (rec(𝐹, 𝐴) ↾ ∅) = ∅
98fveq2i 6893 . . 3 ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ ∅)) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘∅)
10 eqid 2730 . . . 4 (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))
11 simpr 483 . . . . 5 ((𝐴𝑉𝑥 = ∅) → 𝑥 = ∅)
1211iftrued 4535 . . . 4 ((𝐴𝑉𝑥 = ∅) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))) = 𝐴)
13 0ex 5306 . . . . 5 ∅ ∈ V
1413a1i 11 . . . 4 (𝐴𝑉 → ∅ ∈ V)
15 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
1610, 12, 14, 15fvmptd2 7005 . . 3 (𝐴𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘∅) = 𝐴)
179, 16eqtrid 2782 . 2 (𝐴𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐹‘(𝑥 dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ ∅)) = 𝐴)
187, 17eqtrid 2782 1 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  wss 3947  c0 4321  ifcif 4527   cuni 4907  cmpt 5230  dom cdm 5675  ran crn 5676  cres 5677  Lim wlim 6364  cfv 6542  ωcom 7857  reccrdg 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412
This theorem is referenced by: (None)
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