Theorem mptsnunlem 33467

Description: This is the core of the proof of mptsnun 33468, 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 5267	. . . . . . 7 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
2		mptsnun.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
3	2	reseq1i 5535	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵)
4		resmpt 5595	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑥}) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
5	3, 4	syl5eq 2794	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ {𝑥}))
6	5	rneqd 5496	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ {𝑥}))
7		rnmptsn 33464	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
8	6, 7	syl6eq 2798	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → ran (𝐹 ↾ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
9	1, 8	syl5eq 2794	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 “ 𝐵) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
10	9	unieqd 4586	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
11	10	eleq2d 2813	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
12		eleq1 2815	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
13		eluniab 4587	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}))
14		ancom 465	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
15		r19.41v 3215	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢))
16		df-rex 3044	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐵 (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
17	14, 15, 16	3bitr2i 288	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)))
18		eleq2 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = {𝑥} → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
19	18	anbi2d 742	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = {𝑥} → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
20	19	adantr 472	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) ↔ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
21	20	ibi 256	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢) → (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥}))
22	21	anim2i 594	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → (𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
23	22	eximi 1899	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ 𝑢)) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
24	17, 23	sylbi 207	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})))
25		an12 873	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ (𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
26	25	exbii 1911	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑢 = {𝑥} ∧ 𝑧 ∈ {𝑥})) ↔ ∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})))
27		exsimpr 1933	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑢 = {𝑥} ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥})) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
28	26, 27	sylbi 207	. . . . . . . . . . 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 1995	. . . . . . . . 9 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
31	13, 30	sylbi 207	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}))
32		velsn 4325	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
33	32	anbi2i 732	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
34	33	exbii 1911	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ {𝑥}) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
35	31, 34	sylib 208	. . . . . . 7 ⊢ (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥))
36	12	biimparc 505	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 = 𝑥) → 𝑧 ∈ 𝐵)
37	36	exlimiv 1995	. . . . . . 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 3400	. . . . 5 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} → 𝑥 ∈ 𝐵)
40		equid 2082	. . . . . 6 ⊢ 𝑥 = 𝑥
41		eqid 2748	. . . . . . . . . . . 12 ⊢ {𝑥} = {𝑥}
42		snex 5045	. . . . . . . . . . . . . 14 ⊢ {𝑥} ∈ V
43		sbcg 3632	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
44	42, 43	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
45		eqsbc3 3604	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ {𝑥}))
48		df-rex 3044	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}))
49	13	biimpri 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
50	49	19.23bi 2196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}) → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
51	50	expcom 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑥 ∈ 𝐵 𝑢 = {𝑥} → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
52	48, 51	sylbir 225	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
53	52	19.23bi 2196	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ 𝑢 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
54	47, 53	sylbird 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
55	54	sbcth 3579	. . . . . . . . . . . . . . . 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 3606	. . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) → [{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
60		sbcan 3607	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑥 ∈ 𝐵 ∧ 𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}))
61		nfv 1980	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ {𝑥}
62		nfab1 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
63	62	nfuni 4582	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑢∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
64	63	nfcri 2884	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑢 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}
65	61, 64	nfim 1962	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑢(𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
66		sbctt 3629	. . . . . . . . . . . . . . 15 ⊢ (({𝑥} ∈ V ∧ Ⅎ𝑢(𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})) → ([{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})))
67	42, 65, 66	mp2an 710	. . . . . . . . . . . . . 14 ⊢ ([{𝑥} / 𝑢](𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
68	59, 60, 67	3imtr3i 280	. . . . . . . . . . . . 13 ⊢ (([{𝑥} / 𝑢]𝑥 ∈ 𝐵 ∧ [{𝑥} / 𝑢]𝑢 = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
69	44, 46, 68	syl2anbr 498	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ {𝑥} = {𝑥}) → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
70	41, 69	mpan2 709	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (𝑧 ∈ {𝑥} → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
71	32, 70	syl5bir 233	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
72		eleq1 2815	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
73	71, 72	mpbidi 231	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑧 = 𝑥 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
74	73	com12 32	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
75	74	sbimi 2040	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → [𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
76		equsb3 2557	. . . . . . 7 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 ↔ 𝑥 = 𝑥)
77		nfv 1980	. . . . . . . 8 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
78	77	sbf 2505	. . . . . . 7 ⊢ ([𝑥 / 𝑧](𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
79	75, 76, 78	3imtr3i 280	. . . . . 6 ⊢ (𝑥 = 𝑥 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}}))
80	40, 79	ax-mp 5	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
81	39, 80	impbii 199	. . . 4 ⊢ (𝑥 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}} ↔ 𝑥 ∈ 𝐵)
82	11, 81	syl6bb 276	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ ∪ (𝐹 “ 𝐵) ↔ 𝑥 ∈ 𝐵))
83	82	eqrdv 2746	. 2 ⊢ (𝐵 ⊆ 𝐴 → ∪ (𝐹 “ 𝐵) = 𝐵)
84	83	eqcomd 2754	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1620  ∃wex 1841  Ⅎwnf 1845  [wsb 2034   ∈ wcel 2127  {cab 2734  ∃wrex 3039  Vcvv 3328  [wsbc 3564   ⊆ wss 3703  {csn 4309  ∪ cuni 4576   ↦ cmpt 4869  ran crn 5255   ↾ cres 5256   “ cima 5257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-xp 5260  df-rel 5261  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267

This theorem is referenced by:  mptsnun  33468