Theorem mptsnunlem 37590

Description: This is the core of the proof of mptsnun 37591, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)

Hypotheses

Ref	Expression
mptsnun.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
mptsnun.r	⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Assertion

Ref	Expression
mptsnunlem	⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑥,𝐹

Allowed substitution hints:   𝑅(𝑥,𝑢)   𝐹(𝑢)

Proof of Theorem mptsnunlem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5645	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
2		mptsnun.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
3	2	reseq1i 5942	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵)
4		resmpt 6004	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
5	3, 4	eqtrid 2784	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
6	5	rneqd 5895	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ {𝑥}))
7		rnmptsn 37587	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
8	6, 7	eqtrdi 2788	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
9	1, 8	eqtrid 2784	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 “ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
10	9	unieqd 4878	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
11	10	eleq2d 2823	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
12		eleq1w 2820	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
13		eluniab 4879	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}))
14		ancom 460	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
15		r19.41v 3168	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
16		df-rex 3063	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
17	14, 15, 16	3bitr2i 299	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
18		eleq2 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = {𝑥} → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
19	18	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = {𝑥} → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
21	20	ibi 267	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥}))
22	21	anim2i 618	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → (𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
23	22	eximi 1837	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
24	17, 23	sylbi 217	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
25		an12 646	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ (𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
26	25	exbii 1850	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ ∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
27		exsimpr 1871	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
28	26, 27	sylbi 217	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
29	24, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
30	29	exlimiv 1932	. . . . . . . . 9 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
31	13, 30	sylbi 217	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
32		velsn 4598	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
33	32	anbi2i 624	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
34	33	exbii 1850	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
35	31, 34	sylib 218	. . . . . . 7 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
36	12	biimparc 479	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥) → 𝑧 ∈ 𝐵)
37	36	exlimiv 1932	. . . . . . 7 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥) → 𝑧 ∈ 𝐵)
38	35, 37	syl 17	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → 𝑧 ∈ 𝐵)
39	12, 38	vtoclga 3534	. . . . 5 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → 𝑥 ∈ 𝐵)
40		equid 2014	. . . . . 6 ⊢ 𝑥 = 𝑥
41		eqid 2737	. . . . . . . . . . . 12 ⊢ {𝑥} = {𝑥}
42		vsnex 5381	. . . . . . . . . . . . . 14 ⊢ {𝑥} ∈ V
43		sbcg 3815	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
44	42, 43	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
45		eqsbc1 3789	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
46	42, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
47	18	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
48		df-rex 3063	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}))
49	13	biimpri 228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
50	49	19.23bi 2199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
51	50	expcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
52	48, 51	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
53	52	19.23bi 2199	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
54	47, 53	sylbird 260	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
55	54	sbcth 3757	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})))
56	42, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ [{𝑥} / 𝑢]((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
57		sbcimg 3791	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})) ↔ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))))
58	42, 57	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([{𝑥} / 𝑢]((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})) ↔ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})))
59	56, 58	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
60		sbcan 3792	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
61		nfv 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ {𝑥}
62		nfab1 2901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
63	62	nfuni 4872	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑢∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
64	63	nfcri 2891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
65	61, 64	nfim 1898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑢(𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
66	42, 65	sbcgfi 3816	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
67	59, 60, 66	3imtr3i 291	. . . . . . . . . . . . 13 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
68	44, 46, 67	syl2anbr 600	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ {𝑥} = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
69	41, 68	mpan2 692	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
70	32, 69	biimtrrid 243	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
71		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
72	70, 71	mpbidi 241	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
73	72	com12 32	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
74	73	sbimi 2080	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → [𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
75		equsb3 2109	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 ↔ 𝑥 = 𝑥)
76		sbv 2094	. . . . . . 7 ⊢ ([𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
77	74, 75, 76	3imtr3i 291	. . . . . 6 ⊢ (𝑥 = 𝑥 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
78	40, 77	ax-mp 5	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
79	39, 78	impbii 209	. . . 4 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ 𝐵)
80	11, 79	bitrdi 287	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ 𝐵))
81	80	eqrdv 2735	. 2 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = 𝐵)
82	81	eqcomd 2743	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  [wsb 2068   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442  [wsbc 3742   ⊆ wss 3903  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  ran crn 5633   ↾ cres 5634   “ cima 5635

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-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  mptsnun  37591