Theorem tz7.44-2 8338

Description: The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypotheses

Ref	Expression
tz7.44.1	⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))
tz7.44.2	⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))
tz7.44.3	⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V)
tz7.44.4	⊢ 𝐹 Fn 𝑋
tz7.44.5	⊢ Ord 𝑋

Assertion

Ref	Expression
tz7.44-2	⊢ (suc 𝐵 ∈ 𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹‘𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑦,𝐺   𝑥,𝐻   𝑦,𝑋

Allowed substitution hints:   𝐴(𝑦)   𝐺(𝑥)   𝐻(𝑦)   𝑋(𝑥)

Proof of Theorem tz7.44-2

Step	Hyp	Ref	Expression
1		fveq2 6833	. . . 4 ⊢ (𝑦 = suc 𝐵 → (𝐹‘𝑦) = (𝐹‘suc 𝐵))
2		reseq2 5932	. . . . 5 ⊢ (𝑦 = suc 𝐵 → (𝐹 ↾ 𝑦) = (𝐹 ↾ suc 𝐵))
3	2	fveq2d 6837	. . . 4 ⊢ (𝑦 = suc 𝐵 → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
4	1, 3	eqeq12d 2751	. . 3 ⊢ (𝑦 = suc 𝐵 → ((𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)) ↔ (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵))))
5		tz7.44.2	. . 3 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))
6	4, 5	vtoclga 3531	. 2 ⊢ (suc 𝐵 ∈ 𝑋 → (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵)))
7		tz7.44.1	. . 3 ⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))
8		eqeq1 2739	. . . 4 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
9		dmeq 5851	. . . . . 6 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
10		limeq 6328	. . . . . 6 ⊢ (dom 𝑥 = dom (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
11	9, 10	syl 17	. . . . 5 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
12		rneq 5884	. . . . . 6 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
13	12	unieqd 4875	. . . . 5 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → ∪ ran 𝑥 = ∪ ran (𝐹 ↾ suc 𝐵))
14		fveq1 6832	. . . . . . 7 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥‘∪ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘∪ dom 𝑥))
15	9	unieqd 4875	. . . . . . . 8 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → ∪ dom 𝑥 = ∪ dom (𝐹 ↾ suc 𝐵))
16	15	fveq2d 6837	. . . . . . 7 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘∪ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)))
17	14, 16	eqtrd 2770	. . . . . 6 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥‘∪ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)))
18	17	fveq2d 6837	. . . . 5 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥‘∪ dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))
19	11, 13, 18	ifbieq12d 4507	. . . 4 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥))) = if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)))))
20	8, 19	ifbieq2d 4505	. . 3 ⊢ (𝑥 = (𝐹 ↾ suc 𝐵) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))) = if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))))
21	2	eleq1d 2820	. . . 4 ⊢ (𝑦 = suc 𝐵 → ((𝐹 ↾ 𝑦) ∈ V ↔ (𝐹 ↾ suc 𝐵) ∈ V))
22		tz7.44.3	. . . 4 ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V)
23	21, 22	vtoclga 3531	. . 3 ⊢ (suc 𝐵 ∈ 𝑋 → (𝐹 ↾ suc 𝐵) ∈ V)
24		noel 4289	. . . . . . 7 ⊢ ¬ 𝐵 ∈ ∅
25		dmeq 5851	. . . . . . . . 9 ⊢ ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
26		dm0 5868	. . . . . . . . 9 ⊢ dom ∅ = ∅
27	25, 26	eqtrdi 2786	. . . . . . . 8 ⊢ ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
28		tz7.44.5	. . . . . . . . . . . . 13 ⊢ Ord 𝑋
29		ordsson 7728	. . . . . . . . . . . . 13 ⊢ (Ord 𝑋 → 𝑋 ⊆ On)
30	28, 29	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑋 ⊆ On
31		ordtr 6330	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑋 → Tr 𝑋)
32	28, 31	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Tr 𝑋
33		trsuc 6405	. . . . . . . . . . . . 13 ⊢ ((Tr 𝑋 ∧ suc 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
34	32, 33	mpan 691	. . . . . . . . . . . 12 ⊢ (suc 𝐵 ∈ 𝑋 → 𝐵 ∈ 𝑋)
35	30, 34	sselid 3930	. . . . . . . . . . 11 ⊢ (suc 𝐵 ∈ 𝑋 → 𝐵 ∈ On)
36		sucidg 6399	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (suc 𝐵 ∈ 𝑋 → 𝐵 ∈ suc 𝐵)
38		dmres 5970	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
39		ordelss 6332	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ⊆ 𝑋)
40	28, 39	mpan 691	. . . . . . . . . . . . 13 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ⊆ 𝑋)
41		tz7.44.4	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn 𝑋
42	41	fndmi 6595	. . . . . . . . . . . . 13 ⊢ dom 𝐹 = 𝑋
43	40, 42	sseqtrrdi 3974	. . . . . . . . . . . 12 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ⊆ dom 𝐹)
44		dfss2 3918	. . . . . . . . . . . 12 ⊢ (suc 𝐵 ⊆ dom 𝐹 ↔ (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
45	43, 44	sylib 218	. . . . . . . . . . 11 ⊢ (suc 𝐵 ∈ 𝑋 → (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
46	38, 45	eqtrid 2782	. . . . . . . . . 10 ⊢ (suc 𝐵 ∈ 𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
47	37, 46	eleqtrrd 2838	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ 𝑋 → 𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
48		eleq2 2824	. . . . . . . . 9 ⊢ (dom (𝐹 ↾ suc 𝐵) = ∅ → (𝐵 ∈ dom (𝐹 ↾ suc 𝐵) ↔ 𝐵 ∈ ∅))
49	47, 48	syl5ibcom 245	. . . . . . . 8 ⊢ (suc 𝐵 ∈ 𝑋 → (dom (𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
50	27, 49	syl5 34	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → ((𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
51	24, 50	mtoi 199	. . . . . 6 ⊢ (suc 𝐵 ∈ 𝑋 → ¬ (𝐹 ↾ suc 𝐵) = ∅)
52	51	iffalsed 4489	. . . . 5 ⊢ (suc 𝐵 ∈ 𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))) = if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)))))
53		nlimsucg 7784	. . . . . . . 8 ⊢ (𝐵 ∈ On → ¬ Lim suc 𝐵)
54	35, 53	syl 17	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → ¬ Lim suc 𝐵)
55		limeq 6328	. . . . . . . 8 ⊢ (dom (𝐹 ↾ suc 𝐵) = suc 𝐵 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
56	46, 55	syl 17	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
57	54, 56	mtbird 325	. . . . . 6 ⊢ (suc 𝐵 ∈ 𝑋 → ¬ Lim dom (𝐹 ↾ suc 𝐵))
58	57	iffalsed 4489	. . . . 5 ⊢ (suc 𝐵 ∈ 𝑋 → if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)))) = (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))
59	46	unieqd 4875	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ 𝑋 → ∪ dom (𝐹 ↾ suc 𝐵) = ∪ suc 𝐵)
60		eloni 6326	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Ord 𝐵)
61		ordunisuc 7774	. . . . . . . . . 10 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
62	35, 60, 61	3syl 18	. . . . . . . . 9 ⊢ (suc 𝐵 ∈ 𝑋 → ∪ suc 𝐵 = 𝐵)
63	59, 62	eqtrd 2770	. . . . . . . 8 ⊢ (suc 𝐵 ∈ 𝑋 → ∪ dom (𝐹 ↾ suc 𝐵) = 𝐵)
64	63	fveq2d 6837	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → ((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
65	37	fvresd 6853	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹‘𝐵))
66	64, 65	eqtrd 2770	. . . . . 6 ⊢ (suc 𝐵 ∈ 𝑋 → ((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵)) = (𝐹‘𝐵))
67	66	fveq2d 6837	. . . . 5 ⊢ (suc 𝐵 ∈ 𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹‘𝐵)))
68	52, 58, 67	3eqtrd 2774	. . . 4 ⊢ (suc 𝐵 ∈ 𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))) = (𝐻‘(𝐹‘𝐵)))
69		fvex 6846	. . . 4 ⊢ (𝐻‘(𝐹‘𝐵)) ∈ V
70	68, 69	eqeltrdi 2843	. . 3 ⊢ (suc 𝐵 ∈ 𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))) ∈ V)
71	7, 20, 23, 70	fvmptd3 6964	. 2 ⊢ (suc 𝐵 ∈ 𝑋 → (𝐺‘(𝐹 ↾ suc 𝐵)) = if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ∪ ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘∪ dom (𝐹 ↾ suc 𝐵))))))
72	6, 71, 68	3eqtrd 2774	1 ⊢ (suc 𝐵 ∈ 𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3439   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  ifcif 4478  ∪ cuni 4862   ↦ cmpt 5178  Tr wtr 5204  dom cdm 5623  ran crn 5624   ↾ cres 5625  Ord word 6315  Oncon0 6316  Lim wlim 6317  suc csuc 6318   Fn wfn 6486  ‘cfv 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-fv 6499

This theorem is referenced by:  rdgsucg  8354