Theorem dfrdg3 35937

Description: Generalization of dfrdg2 35936 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dfrdg3	⊢ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}

Distinct variable groups:   𝑓,𝐹,𝑥,𝑦   𝑓,𝐼,𝑥,𝑦

Proof of Theorem dfrdg3

Step	Hyp	Ref	Expression
1		dfrdg2 35936	. . 3 ⊢ (𝐼 ∈ V → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
2		iftrue 4483	. . . . . . . . . 10 ⊢ (𝐼 ∈ V → if(𝐼 ∈ V, 𝐼, ∅) = 𝐼)
3	2	ifeq1d 4497	. . . . . . . . 9 ⊢ (𝐼 ∈ V → if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
4	3	eqeq2d 2745	. . . . . . . 8 ⊢ (𝐼 ∈ V → ((𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
5	4	ralbidv 3157	. . . . . . 7 ⊢ (𝐼 ∈ V → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
6	5	anbi2d 630	. . . . . 6 ⊢ (𝐼 ∈ V → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
7	6	rexbidv 3158	. . . . 5 ⊢ (𝐼 ∈ V → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
8	7	abbidv 2800	. . . 4 ⊢ (𝐼 ∈ V → {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
9	8	unieqd 4874	. . 3 ⊢ (𝐼 ∈ V → ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
10	1, 9	eqtr4d 2772	. 2 ⊢ (𝐼 ∈ V → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
11		0ex 5250	. . . 4 ⊢ ∅ ∈ V
12		dfrdg2 35936	. . . 4 ⊢ (∅ ∈ V → rec(𝐹, ∅) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
13	11, 12	ax-mp 5	. . 3 ⊢ rec(𝐹, ∅) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
14		rdgprc 35935	. . 3 ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))
15		iffalse 4486	. . . . . . . . . 10 ⊢ (¬ 𝐼 ∈ V → if(𝐼 ∈ V, 𝐼, ∅) = ∅)
16	15	ifeq1d 4497	. . . . . . . . 9 ⊢ (¬ 𝐼 ∈ V → if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
17	16	eqeq2d 2745	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → ((𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
18	17	ralbidv 3157	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
19	18	anbi2d 630	. . . . . 6 ⊢ (¬ 𝐼 ∈ V → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
20	19	rexbidv 3158	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
21	20	abbidv 2800	. . . 4 ⊢ (¬ 𝐼 ∈ V → {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
22	21	unieqd 4874	. . 3 ⊢ (¬ 𝐼 ∈ V → ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, ∅, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
23	13, 14, 22	3eqtr4a 2795	. 2 ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
24	10, 23	pm2.61i 182	1 ⊢ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049  ∃wrex 3058  Vcvv 3438  ∅c0 4283  ifcif 4477  ∪ cuni 4861   “ cima 5625  Oncon0 6315  Lim wlim 6316   Fn wfn 6485  ‘cfv 6490  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  dfrdg4  36094