Theorem bj-rdg0gALT 37094

Description: Alternate proof of rdg0g 8446. More direct since it bypasses tz7.44-1 8425 and rdg0 8440 (and vtoclg 3538, vtoclga 3561). (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	⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Proof of Theorem bj-rdg0gALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgdmlim 8436	. . . . 5 ⊢ Lim dom rec(𝐹, 𝐴)
2		limomss 7871	. . . . 5 ⊢ (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴))
3	1, 2	ax-mp 5	. . . 4 ⊢ ω ⊆ dom rec(𝐹, 𝐴)
4		peano1 7889	. . . 4 ⊢ ∅ ∈ ω
5	3, 4	sselii 3960	. . 3 ⊢ ∅ ∈ dom rec(𝐹, 𝐴)
6		rdgvalg 8438	. . 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) ↾ ∅)))
7	5, 6	ax-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 5975	. . . 4 ⊢ (rec(𝐹, 𝐴) ↾ ∅) = ∅
9	8	fveq2i 6884	. . 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 2736	. . . 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 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → 𝑥 = ∅)
12	11	iftrued 4513	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))) = 𝐴)
13		0ex 5282	. . . . 5 ⊢ ∅ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
15		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
16	10, 12, 14, 15	fvmptd2 6999	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘∅) = 𝐴)
17	9, 16	eqtrid 2783	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ ∅)) = 𝐴)
18	7, 17	eqtrid 2783	1 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  ifcif 4505  ∪ cuni 4888   ↦ cmpt 5206  dom cdm 5659  ran crn 5660   ↾ cres 5661  Lim wlim 6358  ‘cfv 6536  ωcom 7866  reccrdg 8428

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429

This theorem is referenced by: (None)