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Theorem exrecfnlem 34663

Description: Lemma for exrecfn 34664. (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 8063	. . 3 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2		peano1 7601	. . . 4 ⊢ ∅ ∈ ω
3		omelon 9109	. . . . 5 ⊢ ω ∈ On
4		limom 7595	. . . . 5 ⊢ Lim ω
5		rdglimss 34661	. . . . 5 ⊢ (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
6	3, 4, 5	mpanl12 700	. . . 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 4022	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9		rdglim2a 8069	. . . . . . . 8 ⊢ ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
10	3, 4, 9	mp2an 690	. . . . . . 7 ⊢ (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
11	10	eleq2i 2904	. . . . . 6 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 ∈ ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12		eliun 4923	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
13	11, 12	bitri 277	. . . . 5 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
14		peano2 7602	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15		nnon 7586	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω → 𝑢 ∈ On)
16		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
17	16	elrnmpt1 5830	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18		elun2 4153	. . . . . . . . . . . 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 6683	. . . . . . . . . . . . . 14 ⊢ (rec(𝐹, 𝐴)‘𝑢) ∈ V
21		exrecfnlem.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
22		nfcv 2977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦V
23		nfcv 2977	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦𝑧
24		nfmpt1 5164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑧 ↦ 𝐵)
25	24	nfrn 5824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦ran (𝑦 ∈ 𝑧 ↦ 𝐵)
26	23, 25	nfun 4141	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦(𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))
27	22, 26	nfmpt 5163	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
28	21, 27	nfcxfr 2975	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐹
29		nfcv 2977	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐴
30	28, 29	nfrdg 8050	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦rec(𝐹, 𝐴)
31		nfcv 2977	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦𝑢
32	30, 31	nffv 6680	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘𝑢)
33	32	mptexgf 6985	. . . . . . . . . . . . . . . 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 7617	. . . . . . . . . . . . . 14 ⊢ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
36	20, 35	unex 7469	. . . . . . . . . . . . 13 ⊢ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝐴
38		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑢
39		nfmpt1 5164	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
40	21, 39	nfcxfr 2975	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐹
41	40, 37	nfrdg 8050	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧rec(𝐹, 𝐴)
42	41, 38	nffv 6680	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(rec(𝐹, 𝐴)‘𝑢)
43		nfcv 2977	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐵
44	42, 43	nfmpt 5163	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
45	44	nfrn 5824	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
46	42, 45	nfun 4141	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47		rdgeq1 8047	. . . . . . . . . . . . . . 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 2995	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51		eqidd 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
52	50, 49, 51	mpteq12df 5148	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦 ∈ 𝑧 ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
53	52	rneqd 5808	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦 ∈ 𝑧 ↦ 𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
54	49, 53	uneq12d 4140	. . . . . . . . . . . . . 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 8064	. . . . . . . . . . . . 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 689	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
57	56	eleq2d 2898	. . . . . . . . . . 11 ⊢ (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
58	19, 57	syl5ibr 248	. . . . . . . . . 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 34660	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
61	3, 4, 60	mpanl12 700	. . . . . . . . 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 418	. . . . . . 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 3283	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
66	13, 65	syl5bi 244	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
67	66	alimi 1812	. . 3 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68		df-ral 3143	. . 3 ⊢ (∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
69	67, 68	sylibr 236	. 2 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
70		fvex 6683	. . 3 ⊢ (rec(𝐹, 𝐴)‘ω) ∈ V
71		sseq2 3993	. . . 4 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑦ω
73	30, 72	nffv 6680	. . . . . . 7 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘ω)
74	73	nfeq2 2995	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75		eleq2 2901	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76		eleq2 2901	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
77	75, 76	imbi12d 347	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
78	74, 77	albid 2224	. . . . 5 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
79		df-ral 3143	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥))
80	78, 79, 68	3bitr4g 316	. . . 4 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥 ↔ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
81	71, 80	anbi12d 632	. . 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	70, 81	spcev 3607	. 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, 69, 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 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 ∪ ciun 4919 ↦ cmpt 5146 ran crn 5556 Oncon0 6191 Lim wlim 6192 suc csuc 6193 ‘cfv 6355 ωcom 7580 reccrdg 8045

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046

This theorem is referenced by: exrecfn 34664

W3C validator