Theorem rdgprc 35986

Description: The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rdgprc	⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Proof of Theorem rdgprc

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘∅))
2		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘∅))
3	1, 2	eqeq12d 2752	. . . . . 6 ⊢ (𝑧 = ∅ → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅)))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑧 = ∅ → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))))
5		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑦))
6		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑦))
7	5, 6	eqeq12d 2752	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
8	7	imbi2d 340	. . . . 5 ⊢ (𝑧 = 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦))))
9		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘suc 𝑦))
10		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘suc 𝑦))
11	9, 10	eqeq12d 2752	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
12	11	imbi2d 340	. . . . 5 ⊢ (𝑧 = suc 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
13		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑥))
14		fveq2 6834	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑥))
15	13, 14	eqeq12d 2752	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
16	15	imbi2d 340	. . . . 5 ⊢ (𝑧 = 𝑥 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))))
17		rdgprc0 35985	. . . . . 6 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
18		0ex 5252	. . . . . . 7 ⊢ ∅ ∈ V
19	18	rdg0 8352	. . . . . 6 ⊢ (rec(𝐹, ∅)‘∅) = ∅
20	17, 19	eqtr4di 2789	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))
21		fveq2 6834	. . . . . . 7 ⊢ ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
22		rdgsuc 8355	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, 𝐼)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
23		rdgsuc 8355	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, ∅)‘suc 𝑦) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
24	22, 23	eqeq12d 2752	. . . . . . 7 ⊢ (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦))))
25	21, 24	imbitrrid 246	. . . . . 6 ⊢ (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
26	25	imim2d 57	. . . . 5 ⊢ (𝑦 ∈ On → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
27		r19.21v 3161	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑧 (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) ↔ (¬ 𝐼 ∈ V → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
28		limord 6378	. . . . . . . . 9 ⊢ (Lim 𝑧 → Ord 𝑧)
29		ordsson 7728	. . . . . . . . 9 ⊢ (Ord 𝑧 → 𝑧 ⊆ On)
30		rdgfnon 8349	. . . . . . . . . 10 ⊢ rec(𝐹, 𝐼) Fn On
31		rdgfnon 8349	. . . . . . . . . 10 ⊢ rec(𝐹, ∅) Fn On
32		fvreseq 6985	. . . . . . . . . 10 ⊢ (((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) ∧ 𝑧 ⊆ On) → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
33	30, 31, 32	mpanl12 702	. . . . . . . . 9 ⊢ (𝑧 ⊆ On → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
34	28, 29, 33	3syl 18	. . . . . . . 8 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
35		rneq 5885	. . . . . . . . . . 11 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ran (rec(𝐹, 𝐼) ↾ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧))
36		df-ima 5637	. . . . . . . . . . 11 ⊢ (rec(𝐹, 𝐼) “ 𝑧) = ran (rec(𝐹, 𝐼) ↾ 𝑧)
37		df-ima 5637	. . . . . . . . . . 11 ⊢ (rec(𝐹, ∅) “ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧)
38	35, 36, 37	3eqtr4g 2796	. . . . . . . . . 10 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
39	38	unieqd 4876	. . . . . . . . 9 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
40		vex 3444	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
41		rdglim 8357	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = ∪ (rec(𝐹, 𝐼) “ 𝑧))
42		rdglim 8357	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, ∅)‘𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
43	41, 42	eqeq12d 2752	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
44	40, 43	mpan 690	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
45	39, 44	imbitrrid 246	. . . . . . . 8 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
46	34, 45	sylbird 260	. . . . . . 7 ⊢ (Lim 𝑧 → (∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
47	46	imim2d 57	. . . . . 6 ⊢ (Lim 𝑧 → ((¬ 𝐼 ∈ V → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
48	27, 47	biimtrid 242	. . . . 5 ⊢ (Lim 𝑧 → (∀𝑦 ∈ 𝑧 (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
49	4, 8, 12, 16, 20, 26, 48	tfinds 7802	. . . 4 ⊢ (𝑥 ∈ On → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
50	49	com12 32	. . 3 ⊢ (¬ 𝐼 ∈ V → (𝑥 ∈ On → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
51	50	ralrimiv 3127	. 2 ⊢ (¬ 𝐼 ∈ V → ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
52		eqfnfv 6976	. . 3 ⊢ ((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) → (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
53	30, 31, 52	mp2an 692	. 2 ⊢ (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
54	51, 53	sylibr 234	1 ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  ∪ cuni 4863  ran crn 5625   ↾ cres 5626   “ cima 5627  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319   Fn wfn 6487  ‘cfv 6492  reccrdg 8340

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  dfrdg3  35988