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Theorem exrecfnlem 37361

Description: Lemma for exrecfn 37362. (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 8465	. . 3 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2		peano1 7910	. . . 4 ⊢ ∅ ∈ ω
3		omelon 9683	. . . . 5 ⊢ ω ∈ On
4		limom 7902	. . . . 5 ⊢ Lim ω
5		rdglimss 37359	. . . . 5 ⊢ (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
6	3, 4, 5	mpanl12 702	. . . 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 4050	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9		rdglim2a 8471	. . . . . . . 8 ⊢ ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
10	3, 4, 9	mp2an 692	. . . . . . 7 ⊢ (rec(𝐹, 𝐴)‘ω) = ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
11	10	eleq2i 2830	. . . . . 6 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 ∈ ∪ 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12		eliun 4999	. . . . . 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 7912	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15		nnon 7892	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω → 𝑢 ∈ On)
16		eqid 2734	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
17	16	elrnmpt1 5973	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18		elun2 4192	. . . . . . . . . . . 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 6919	. . . . . . . . . . . . . 14 ⊢ (rec(𝐹, 𝐴)‘𝑢) ∈ V
21		exrecfnlem.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
22		nfcv 2902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦V
23		nfcv 2902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦𝑧
24		nfmpt1 5255	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑧 ↦ 𝐵)
25	24	nfrn 5965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦ran (𝑦 ∈ 𝑧 ↦ 𝐵)
26	23, 25	nfun 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑦(𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))
27	22, 26	nfmpt 5254	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
28	21, 27	nfcxfr 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐹
29		nfcv 2902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦𝐴
30	28, 29	nfrdg 8452	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦rec(𝐹, 𝐴)
31		nfcv 2902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦𝑢
32	30, 31	nffv 6916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘𝑢)
33	32	mptexgf 7241	. . . . . . . . . . . . . . . 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 7932	. . . . . . . . . . . . . 14 ⊢ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
36	20, 35	unex 7762	. . . . . . . . . . . . 13 ⊢ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37		nfcv 2902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝐴
38		nfcv 2902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧𝑢
39		nfmpt1 5255	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))
40	21, 39	nfcxfr 2900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐹
41	40, 37	nfrdg 8452	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧rec(𝐹, 𝐴)
42	41, 38	nffv 6916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(rec(𝐹, 𝐴)‘𝑢)
43		nfcv 2902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑧𝐵
44	42, 43	nfmpt 5254	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
45	44	nfrn 5965	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
46	42, 45	nfun 4179	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47		rdgeq1 8449	. . . . . . . . . . . . . . 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 2920	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51		eqidd 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
52	50, 49, 51	mpteq12df 5233	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦 ∈ 𝑧 ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
53	52	rneqd 5951	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦 ∈ 𝑧 ↦ 𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
54	49, 53	uneq12d 4178	. . . . . . . . . . . . . 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 8466	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
57	56	eleq2d 2824	. . . . . . . . . . 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 37358	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
61	3, 4, 60	mpanl12 702	. . . . . . . . 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 3150	. . . . 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 1807	. . 3 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68		df-ral 3059	. . 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 234	. 2 ⊢ (∀𝑦 𝐵 ∈ 𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
70		fvex 6919	. . 3 ⊢ (rec(𝐹, 𝐴)‘ω) ∈ V
71		sseq2 4021	. . . 4 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑦ω
73	30, 72	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑦(rec(𝐹, 𝐴)‘ω)
74	73	nfeq2 2920	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75		eleq2 2827	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76		eleq2 2827	. . . . . . 7 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵 ∈ 𝑥 ↔ 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
77	75, 76	imbi12d 344	. . . . . 6 ⊢ (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
78	74, 77	albid 2219	. . . . 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 3059	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝐵 ∈ 𝑥))
80	78, 79, 68	3bitr4g 314	. . . 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 3605	. 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 596	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 1534 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 ∪ ciun 4995 ↦ cmpt 5230 ran crn 5689 Oncon0 6385 Lim wlim 6386 suc csuc 6387 ‘cfv 6562 ωcom 7886 reccrdg 8447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-inf2 9678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448

This theorem is referenced by: exrecfn 37362

W3C validator