Theorem mptsnunlem 36822

Description: This is the core of the proof of mptsnun 36823, 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 5693	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
2		mptsnun.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
3	2	reseq1i 5983	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵)
4		resmpt 6044	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
5	3, 4	eqtrid 2779	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
6	5	rneqd 5942	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ {𝑥}))
7		rnmptsn 36819	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
8	6, 7	eqtrdi 2783	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
9	1, 8	eqtrid 2779	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 “ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
10	9	unieqd 4923	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
11	10	eleq2d 2814	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
12		eleq1w 2811	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
13		eluniab 4924	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}))
14		ancom 459	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
15		r19.41v 3184	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
16		df-rex 3067	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
17	14, 15, 16	3bitr2i 298	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
18		eleq2 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = {𝑥} → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
19	18	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = {𝑥} → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
20	19	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
21	20	ibi 266	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥}))
22	21	anim2i 615	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → (𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
23	22	eximi 1829	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
24	17, 23	sylbi 216	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
25		an12 643	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ (𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
26	25	exbii 1842	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ ∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
27		exsimpr 1864	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
28	26, 27	sylbi 216	. . . . . . . . . . 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 1925	. . . . . . . . 9 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
31	13, 30	sylbi 216	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
32		velsn 4646	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
33	32	anbi2i 621	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
34	33	exbii 1842	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
35	31, 34	sylib 217	. . . . . . 7 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
36	12	biimparc 478	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥) → 𝑧 ∈ 𝐵)
37	36	exlimiv 1925	. . . . . . 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 3563	. . . . 5 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → 𝑥 ∈ 𝐵)
40		equid 2007	. . . . . 6 ⊢ 𝑥 = 𝑥
41		eqid 2727	. . . . . . . . . . . 12 ⊢ {𝑥} = {𝑥}
42		vsnex 5433	. . . . . . . . . . . . . 14 ⊢ {𝑥} ∈ V
43		sbcg 3855	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
44	42, 43	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
45		eqsbc1 3826	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
48		df-rex 3067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}))
49	13	biimpri 227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
50	49	19.23bi 2179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
51	50	expcom 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
52	48, 51	sylbir 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
53	52	19.23bi 2179	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
54	47, 53	sylbird 259	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
55	54	sbcth 3791	. . . . . . . . . . . . . . . 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 3828	. . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
60		sbcan 3829	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
61		nfv 1909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ {𝑥}
62		nfab1 2900	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
63	62	nfuni 4917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑢∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
64	63	nfcri 2885	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
65	61, 64	nfim 1891	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑢(𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
66	42, 65	sbcgfi 3857	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
67	59, 60, 66	3imtr3i 290	. . . . . . . . . . . . 13 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
68	44, 46, 67	syl2anbr 597	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ {𝑥} = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
69	41, 68	mpan2 689	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
70	32, 69	biimtrrid 242	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
71		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
72	70, 71	mpbidi 240	. . . . . . . . 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 2069	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → [𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
75		equsb3 2093	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 ↔ 𝑥 = 𝑥)
76		sbv 2083	. . . . . . 7 ⊢ ([𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
77	74, 75, 76	3imtr3i 290	. . . . . 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 208	. . . 4 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ 𝐵)
80	11, 79	bitrdi 286	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ 𝐵))
81	80	eqrdv 2725	. 2 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = 𝐵)
82	81	eqcomd 2733	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773  [wsb 2059   ∈ wcel 2098  {cab 2704  ∃wrex 3066  Vcvv 3471  [wsbc 3776   ⊆ wss 3947  {csn 4630  ∪ cuni 4910   ↦ cmpt 5233  ran crn 5681   ↾ cres 5682   “ cima 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693

This theorem is referenced by:  mptsnun  36823