Theorem bj-rdg0gALT 35251

Description: Alternate proof of rdg0g 8250. More direct since it bypasses tz7.44-1 8229 and rdg0 8244 (and vtoclg 3504, vtoclga 3512). (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 8240	. . . . 5 ⊢ Lim dom rec(𝐹, 𝐴)
2		limomss 7712	. . . . 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 7730	. . . 4 ⊢ ∅ ∈ ω
5	3, 4	sselii 3923	. . 3 ⊢ ∅ ∈ dom rec(𝐹, 𝐴)
6		rdgvalg 8242	. . 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 5894	. . . 4 ⊢ (rec(𝐹, 𝐴) ↾ ∅) = ∅
9	8	fveq2i 6774	. . 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 2740	. . . 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 485	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → 𝑥 = ∅)
12	11	iftrued 4473	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 = ∅) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))) = 𝐴)
13		0ex 5235	. . . . 5 ⊢ ∅ ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
15		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
16	10, 12, 14, 15	fvmptd2 6880	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘∅) = 𝐴)
17	9, 16	eqtrid 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	eqtrid 2792	1 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ⊆ wss 3892  ∅c0 4262  ifcif 4465  ∪ cuni 4845   ↦ cmpt 5162  dom cdm 5590  ran crn 5591   ↾ cres 5592  Lim wlim 6266  ‘cfv 6432  ωcom 7707  reccrdg 8232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7275  df-om 7708  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233

This theorem is referenced by: (None)