Theorem tz7.44-3 8333

Description: The value of 𝐹 at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

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
tz7.44-3	⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = ∪ (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 tz7.44-3

Step	Hyp	Ref	Expression
1		fveq2 6828	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
2		reseq2 5927	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹 ↾ 𝑦) = (𝐹 ↾ 𝐵))
3	2	fveq2d 6832	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘(𝐹 ↾ 𝐵)))
4	1, 3	eqeq12d 2749	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)) ↔ (𝐹‘𝐵) = (𝐺‘(𝐹 ↾ 𝐵))))
5		tz7.44.2	. . . . 5 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))
6	4, 5	vtoclga 3529	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (𝐹‘𝐵) = (𝐺‘(𝐹 ↾ 𝐵)))
7	6	adantr 480	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = (𝐺‘(𝐹 ↾ 𝐵)))
8	2	eleq1d 2818	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐹 ↾ 𝑦) ∈ V ↔ (𝐹 ↾ 𝐵) ∈ V))
9		tz7.44.3	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V)
10	8, 9	vtoclga 3529	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → (𝐹 ↾ 𝐵) ∈ V)
11	10	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹 ↾ 𝐵) ∈ V)
12		simpr 484	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → Lim 𝐵)
13		nlim0 6371	. . . . . . . . . . 11 ⊢ ¬ Lim ∅
14		dmres 5965	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
15		tz7.44.5	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 𝑋
16		ordelss 6327	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ⊆ 𝑋)
17	15, 16	mpan 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ⊆ 𝑋)
18	17	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ 𝑋)
19		tz7.44.4	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn 𝑋
20		fndm 6589	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
21	19, 20	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝐹 = 𝑋
22	18, 21	sseqtrrdi 3972	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ dom 𝐹)
23		dfss2 3916	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
24	22, 23	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐵 ∩ dom 𝐹) = 𝐵)
25	14, 24	eqtrid 2780	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → dom (𝐹 ↾ 𝐵) = 𝐵)
26		dmeq 5847	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ 𝐵) = ∅ → dom (𝐹 ↾ 𝐵) = dom ∅)
27		dm0 5864	. . . . . . . . . . . . . 14 ⊢ dom ∅ = ∅
28	26, 27	eqtrdi 2784	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ 𝐵) = ∅ → dom (𝐹 ↾ 𝐵) = ∅)
29	25, 28	sylan9req 2789	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑋 ∧ Lim 𝐵) ∧ (𝐹 ↾ 𝐵) = ∅) → 𝐵 = ∅)
30		limeq 6323	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑋 ∧ Lim 𝐵) ∧ (𝐹 ↾ 𝐵) = ∅) → (Lim 𝐵 ↔ Lim ∅))
32	13, 31	mtbiri 327	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑋 ∧ Lim 𝐵) ∧ (𝐹 ↾ 𝐵) = ∅) → ¬ Lim 𝐵)
33	32	ex 412	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → ((𝐹 ↾ 𝐵) = ∅ → ¬ Lim 𝐵))
34	12, 33	mt2d 136	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → ¬ (𝐹 ↾ 𝐵) = ∅)
35	34	iffalsed 4485	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))) = if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵)))))
36		limeq 6323	. . . . . . . . . 10 ⊢ (dom (𝐹 ↾ 𝐵) = 𝐵 → (Lim dom (𝐹 ↾ 𝐵) ↔ Lim 𝐵))
37	25, 36	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (Lim dom (𝐹 ↾ 𝐵) ↔ Lim 𝐵))
38	12, 37	mpbird 257	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → Lim dom (𝐹 ↾ 𝐵))
39	38	iftrued 4482	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵)))) = ∪ ran (𝐹 ↾ 𝐵))
40	35, 39	eqtrd 2768	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))) = ∪ ran (𝐹 ↾ 𝐵))
41		rnexg 7838	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐵) ∈ V → ran (𝐹 ↾ 𝐵) ∈ V)
42		uniexg 7679	. . . . . . 7 ⊢ (ran (𝐹 ↾ 𝐵) ∈ V → ∪ ran (𝐹 ↾ 𝐵) ∈ V)
43	11, 41, 42	3syl 18	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → ∪ ran (𝐹 ↾ 𝐵) ∈ V)
44	40, 43	eqeltrd 2833	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))) ∈ V)
45		eqeq1 2737	. . . . . . 7 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))
46		dmeq 5847	. . . . . . . . 9 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → dom 𝑥 = dom (𝐹 ↾ 𝐵))
47		limeq 6323	. . . . . . . . 9 ⊢ (dom 𝑥 = dom (𝐹 ↾ 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ 𝐵)))
48	46, 47	syl 17	. . . . . . . 8 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ 𝐵)))
49		rneq 5880	. . . . . . . . 9 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → ran 𝑥 = ran (𝐹 ↾ 𝐵))
50	49	unieqd 4871	. . . . . . . 8 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → ∪ ran 𝑥 = ∪ ran (𝐹 ↾ 𝐵))
51		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → (𝑥‘∪ dom 𝑥) = ((𝐹 ↾ 𝐵)‘∪ dom 𝑥))
52	46	unieqd 4871	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → ∪ dom 𝑥 = ∪ dom (𝐹 ↾ 𝐵))
53	52	fveq2d 6832	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘∪ dom 𝑥) = ((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵)))
54	51, 53	eqtrd 2768	. . . . . . . . 9 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → (𝑥‘∪ dom 𝑥) = ((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵)))
55	54	fveq2d 6832	. . . . . . . 8 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → (𝐻‘(𝑥‘∪ dom 𝑥)) = (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))
56	48, 50, 55	ifbieq12d 4503	. . . . . . 7 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥))) = if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵)))))
57	45, 56	ifbieq2d 4501	. . . . . 6 ⊢ (𝑥 = (𝐹 ↾ 𝐵) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))) = if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))))
58		tz7.44.1	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))
59	57, 58	fvmptg 6933	. . . . 5 ⊢ (((𝐹 ↾ 𝐵) ∈ V ∧ if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))) ∈ V) → (𝐺‘(𝐹 ↾ 𝐵)) = if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))))
60	11, 44, 59	syl2anc 584	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹 ↾ 𝐵)) = if((𝐹 ↾ 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ 𝐵), ∪ ran (𝐹 ↾ 𝐵), (𝐻‘((𝐹 ↾ 𝐵)‘∪ dom (𝐹 ↾ 𝐵))))))
61	60, 40	eqtrd 2768	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹 ↾ 𝐵)) = ∪ ran (𝐹 ↾ 𝐵))
62	7, 61	eqtrd 2768	. 2 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = ∪ ran (𝐹 ↾ 𝐵))
63		df-ima 5632	. . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
64	63	unieqi 4870	. 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ran (𝐹 ↾ 𝐵)
65	62, 64	eqtr4di 2786	1 ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = ∪ (𝐹 “ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  ifcif 4474  ∪ cuni 4858   ↦ cmpt 5174  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Ord word 6310  Lim wlim 6312   Fn wfn 6481  ‘cfv 6486

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 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-lim 6316  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  rdglimg  8350