Theorem bj-rdg0gALT 35145

Description: Alternate proof of rdg0g 8205. More direct since it bypasses tz7.44-1 8184 and rdg0 8199 (and vtoclg 3496, vtoclga 3504). (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 8195	. . . . 5 ⊢ Lim dom rec(𝐹, 𝐴)
2		limomss 7689	. . . . 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 7707	. . . 4 ⊢ ∅ ∈ ω
5	3, 4	sselii 3915	. . 3 ⊢ ∅ ∈ dom rec(𝐹, 𝐴)
6		rdgvalg 8197	. . 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 5883	. . . 4 ⊢ (rec(𝐹, 𝐴) ↾ ∅) = ∅
9	8	fveq2i 6756	. . 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 2739	. . . 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 488	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → 𝑥 = ∅)
12	11	iftrued 4464	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))) = 𝐴)
13		0ex 5224	. . . . 5 ⊢ ∅ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
15		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
16	10, 12, 14, 15	fvmptd2 6862	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘∅) = 𝐴)
17	9, 16	syl5eq 2792	. 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	syl5eq 2792	1 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  Vcvv 3423   ⊆ wss 3884  ∅c0 4254  ifcif 4456  ∪ cuni 4836   ↦ cmpt 5152  dom cdm 5579  ran crn 5580   ↾ cres 5581  Lim wlim 6249  ‘cfv 6415  ωcom 7684  reccrdg 8187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346  ax-un 7563

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3713  df-csb 3830  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5186  df-id 5479  df-eprel 5485  df-po 5493  df-so 5494  df-fr 5534  df-we 5536  df-xp 5585  df-rel 5586  df-cnv 5587  df-co 5588  df-dm 5589  df-rn 5590  df-res 5591  df-ima 5592  df-pred 6189  df-ord 6251  df-on 6252  df-lim 6253  df-suc 6254  df-iota 6373  df-fun 6417  df-fn 6418  df-f 6419  df-f1 6420  df-fo 6421  df-f1o 6422  df-fv 6423  df-om 7685  df-wrecs 8089  df-recs 8150  df-rdg 8188

This theorem is referenced by: (None)