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Theorem exrecfnlem 37712

Description: Lemma for exrecfn 37713. (Contributed by ML, 30-Mar-2022.)

**Hypothesis**
Ref	Expression
exrecfnlem.1	⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))

**Assertion**
Ref	Expression
exrecfnlem	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))

Distinct variable groups: 𝑦,𝐴,𝑧,𝑥 𝑥,𝐵,𝑧 𝑥,𝐹 𝑦,𝑊 Allowed substitution hints: 𝐵(𝑦) 𝐹(𝑦,𝑧) 𝑉(𝑥,𝑦,𝑧) 𝑊(𝑥,𝑧)

Proof of Theorem exrecfnlem Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdg0g 8360	. . 3 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2		peano1 7834	. . . 4 ⊢ ∅ ∈ ω
3		omelon 9561	. . . . 5 ⊢ ω ∈ On
4		limom 7827	. . . . 5 ⊢ Lim ω
5		rdglimss 37710	. . . . 5 ⊢ (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
6	3, 4, 5	mpanl12 703	. . . 4 ⊢ (∅ ∈ ω → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
7	2, 6	ax-mp 5	. . 3 ⊢ (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω)
8	1, 7	eqsstrrdi 3968	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9		rdglim2a 8366	. . . . . . . 8 ⊢ ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
10	3, 4, 9	mp2an 693	. . . . . . 7 ⊢ (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
11	10	eleq2i 2829	. . . . . 6 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 ∈ ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12		eliun 4938	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
13	11, 12	bitri 275	. . . . 5 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
14		peano2 7835	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15		nnon 7817	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω → 𝑢 ∈ On)
16		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
17	16	elrnmpt1 5910	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18		elun2 4124	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) → 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
20		fvex 6848	. . . . . . . . . . . . . 14 ⊢ (rec(𝐹, 𝐴)‘𝑢) ∈ V
21		exrecfnlem.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
22		nfcv 2899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦V
23		nfcv 2899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦𝑧
24		nfmpt1 5185	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑧 ↦ 𝐵)
25	24	nfrn 5902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦ran (𝑦 ∈ 𝑧 ↦ 𝐵)
26	23, 25	nfun 4111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦(𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))
27	22, 26	nfmpt 5184	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
28	21, 27	nfcxfr 2897	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐹
29		nfcv 2899	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐴
30	28, 29	nfrdg 8347	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦rec(𝐹, 𝐴)
31		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦𝑢
32	30, 31	nffv 6845	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘𝑢)
33	32	mptexgf 7171	. . . . . . . . . . . . . . . 16 ⊢ ((rec(𝐹, 𝐴)‘𝑢) ∈ V → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V)
34	20, 33	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
35	34	rnex 7855	. . . . . . . . . . . . . 14 ⊢ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
36	20, 35	unex 7692	. . . . . . . . . . . . 13 ⊢ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝐴
38		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑢
39		nfmpt1 5185	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
40	21, 39	nfcxfr 2897	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐹
41	40, 37	nfrdg 8347	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧rec(𝐹, 𝐴)
42	41, 38	nffv 6845	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(rec(𝐹, 𝐴)‘𝑢)
43		nfcv 2899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐵
44	42, 43	nfmpt 5184	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
45	44	nfrn 5902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
46	42, 45	nfun 4111	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47		rdgeq1 8344	. . . . . . . . . . . . . . 15 ⊢ (𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))) → rec(𝐹, 𝐴) = rec((𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))), 𝐴))
48	21, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ rec(𝐹, 𝐴) = rec((𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))), 𝐴)
49		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝑧 = (rec(𝐹, 𝐴)‘𝑢))
50	32	nfeq2 2917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51		eqidd 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
52	50, 49, 51	mpteq12df 5170	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦 ∈ 𝑧 ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
53	52	rneqd 5888	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦 ∈ 𝑧 ↦ 𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
54	49, 53	uneq12d 4110	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
55	37, 38, 46, 48, 54	rdgsucmptf 8361	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ On ∧ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V) → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
56	36, 55	mpan2 692	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
57	56	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
58	19, 57	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝑢 ∈ On → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢)))
59	15, 58	syl 17	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢)))
60		rdgellim 37709	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
61	3, 4, 60	mpanl12 703	. . . . . . . . 9 ⊢ (suc 𝑢 ∈ ω → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
62	14, 59, 61	sylsyld 61	. . . . . . . 8 ⊢ (𝑢 ∈ ω → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
63	62	expd 415	. . . . . . 7 ⊢ (𝑢 ∈ ω → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → (𝐵 ∈ 𝑊 → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
64	63	com3r 87	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → (𝑢 ∈ ω → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
65	64	rexlimdv 3137	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
66	13, 65	biimtrid 242	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
67	66	alimi 1813	. . 3 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68	67	ralrid 3060	. 2 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
69		fvex 6848	. . 3 ⊢ (rec(𝐹, 𝐴)‘ω) ∈ V
70		sseq2 3949	. . . 4 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
71		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑦ω
72	30, 71	nffv 6845	. . . . . . 7 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘ω)
73	72	nfeq2 2917	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
74		eleq2 2826	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
75		eleq2 2826	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
76	74, 75	imbi12d 344	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
77	73, 76	albid 2230	. . . . 5 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
78		df-ral 3053	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥))
79		df-ral 3053	. . . . 5 ⊢ (∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
80	77, 78, 79	3bitr4g 314	. . . 4 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥 ↔ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
81	70, 80	anbi12d 633	. . 3 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥) ↔ (𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
82	69, 81	spcev 3549	. 2 ⊢ ((𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))
83	8, 68, 82	syl2an 597	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∀wal 1540 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ∪ cun 3888 ⊆ wss 3890 ∅c0 4274 ∪ ciun 4934 ↦ cmpt 5167 ran crn 5626 Oncon0 6318 Lim wlim 6319 suc csuc 6320 ‘cfv 6493 ωcom 7811 reccrdg 8342

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 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pr 5371 ax-un 7683 ax-inf2 9556

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343

This theorem is referenced by: exrecfn 37713

W3C validator