Theorem rdgprc 31700

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 6191	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘∅))
2		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘∅))
3	1, 2	eqeq12d 2637	. . . . . 6 ⊢ (𝑧 = ∅ → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅)))
4	3	imbi2d 330	. . . . 5 ⊢ (𝑧 = ∅ → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))))
5		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑦))
6		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑦))
7	5, 6	eqeq12d 2637	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
8	7	imbi2d 330	. . . . 5 ⊢ (𝑧 = 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦))))
9		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘suc 𝑦))
10		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘suc 𝑦))
11	9, 10	eqeq12d 2637	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
12	11	imbi2d 330	. . . . 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 6191	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑥))
14		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑥))
15	13, 14	eqeq12d 2637	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
16	15	imbi2d 330	. . . . 5 ⊢ (𝑧 = 𝑥 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))))
17		rdgprc0 31699	. . . . . 6 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
18		0ex 4790	. . . . . . 7 ⊢ ∅ ∈ V
19	18	rdg0 7517	. . . . . 6 ⊢ (rec(𝐹, ∅)‘∅) = ∅
20	17, 19	syl6eqr 2674	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))
21		fveq2 6191	. . . . . . 7 ⊢ ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
22		rdgsuc 7520	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, 𝐼)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
23		rdgsuc 7520	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, ∅)‘suc 𝑦) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
24	22, 23	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦))))
25	21, 24	syl5ibr 236	. . . . . 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 2960	. . . . . 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 5784	. . . . . . . . 9 ⊢ (Lim 𝑧 → Ord 𝑧)
29		ordsson 6989	. . . . . . . . 9 ⊢ (Ord 𝑧 → 𝑧 ⊆ On)
30		rdgfnon 7514	. . . . . . . . . 10 ⊢ rec(𝐹, 𝐼) Fn On
31		rdgfnon 7514	. . . . . . . . . 10 ⊢ rec(𝐹, ∅) Fn On
32		fvreseq 6319	. . . . . . . . . 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 718	. . . . . . . . 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 5351	. . . . . . . . . . 11 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ran (rec(𝐹, 𝐼) ↾ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧))
36		df-ima 5127	. . . . . . . . . . 11 ⊢ (rec(𝐹, 𝐼) “ 𝑧) = ran (rec(𝐹, 𝐼) ↾ 𝑧)
37		df-ima 5127	. . . . . . . . . . 11 ⊢ (rec(𝐹, ∅) “ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧)
38	35, 36, 37	3eqtr4g 2681	. . . . . . . . . 10 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
39	38	unieqd 4446	. . . . . . . . 9 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
40		vex 3203	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
41		rdglim 7522	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = ∪ (rec(𝐹, 𝐼) “ 𝑧))
42		rdglim 7522	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, ∅)‘𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
43	41, 42	eqeq12d 2637	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
44	40, 43	mpan 706	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
45	39, 44	syl5ibr 236	. . . . . . . 8 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
46	34, 45	sylbird 250	. . . . . . 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	syl5bi 232	. . . . 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 7059	. . . 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 2965	. 2 ⊢ (¬ 𝐼 ∈ V → ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
52		eqfnfv 6311	. . 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 708	. 2 ⊢ (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
54	51, 53	sylibr 224	1 ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ran crn 5115   ↾ cres 5116   “ cima 5117  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  dfrdg3  31702