Theorem resf1o 31986

Description: Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)

Hypotheses

Ref	Expression
resf1o.1	⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
resf1o.2	⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶))

Assertion

Ref	Expression
resf1o	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑍

Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem resf1o

Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resf1o.2	. 2 ⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶))
2		resexg 6028	. . 3 ⊢ (𝑓 ∈ 𝑋 → (𝑓 ↾ 𝐶) ∈ V)
3	2	adantl 483	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ 𝐶) ∈ V)
4		simpr 486	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → 𝑔 ∈ (𝐵 ↑m 𝐶))
5		difexg 5328	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐶) ∈ V)
6	5	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∖ 𝐶) ∈ V)
7		snex 5432	. . . . . 6 ⊢ {𝑍} ∈ V
8		xpexg 7737	. . . . . 6 ⊢ (((𝐴 ∖ 𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
9	6, 7, 8	sylancl 587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
10	9	adantr 482	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
11		unexg 7736	. . . 4 ⊢ ((𝑔 ∈ (𝐵 ↑m 𝐶) ∧ ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
12	4, 10, 11	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
13	12	adantlr 714	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
14		resf1o.1	. . . . 5 ⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
15	14	reqabi 3455	. . . 4 ⊢ (𝑓 ∈ 𝑋 ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
16	15	anbi1i 625	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 = (𝑓 ↾ 𝐶)) ↔ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶)))
17		simprr 772	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑔 = (𝑓 ↾ 𝐶))
18		simprll 778	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑓 ∈ (𝐵 ↑m 𝐴))
19		elmapi 8843	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝐵)
20	18, 19	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑓:𝐴⟶𝐵)
21		simp3 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
22	21	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝐶 ⊆ 𝐴)
23	20, 22	fssresd 6759	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ 𝐶):𝐶⟶𝐵)
24		simp2 1138	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ 𝑊)
25		simp1 1137	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ 𝑉)
26	25, 21	ssexd 5325	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V)
27		elmapg 8833	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ V) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
28	24, 26, 27	syl2anc 585	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
29	28	ad2antrr 725	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
30	23, 29	mpbird 257	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶))
31	17, 30	eqeltrd 2834	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑔 ∈ (𝐵 ↑m 𝐶))
32		undif 4482	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
33	32	biimpi 215	. . . . . . . . . 10 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
34	33	reseq2d 5982	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))) = (𝑓 ↾ 𝐴))
35	22, 34	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))) = (𝑓 ↾ 𝐴))
36		ffn 6718	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
37		fnresdm 6670	. . . . . . . . 9 ⊢ (𝑓 Fn 𝐴 → (𝑓 ↾ 𝐴) = 𝑓)
38	20, 36, 37	3syl 18	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ 𝐴) = 𝑓)
39	35, 38	eqtr2d 2774	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))))
40		resundi 5996	. . . . . . 7 ⊢ (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))) = ((𝑓 ↾ 𝐶) ∪ (𝑓 ↾ (𝐴 ∖ 𝐶)))
41	39, 40	eqtrdi 2789	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑓 = ((𝑓 ↾ 𝐶) ∪ (𝑓 ↾ (𝐴 ∖ 𝐶))))
42	17	eqcomd 2739	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ 𝐶) = 𝑔)
43		simprlr 779	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
44	25	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝐴 ∈ 𝑉)
45		simplr 768	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑍 ∈ 𝐵)
46		eqid 2733	. . . . . . . . . . 11 ⊢ (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
47	46	ffs2 31984	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵 ∧ 𝑓:𝐴⟶𝐵) → (𝑓 supp 𝑍) = (◡𝑓 “ (𝐵 ∖ {𝑍})))
48	44, 45, 20, 47	syl3anc 1372	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 supp 𝑍) = (◡𝑓 “ (𝐵 ∖ {𝑍})))
49		sseqin2 4216	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
50	49	biimpi 215	. . . . . . . . . 10 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
51	22, 50	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝐴 ∩ 𝐶) = 𝐶)
52	43, 48, 51	3sstr4d 4030	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶))
53		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 ∈ (𝐵 ↑m 𝐴))
54	53, 19, 36	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 Fn 𝐴)
55		inundif 4479	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
56	55	fneq2i 6648	. . . . . . . . . . 11 ⊢ (𝑓 Fn ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) ↔ 𝑓 Fn 𝐴)
57	54, 56	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 Fn ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)))
58		vex 3479	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 ∈ V)
60		simpr 486	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
61		inindif 31785	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
63		fnsuppres 8176	. . . . . . . . . 10 ⊢ ((𝑓 Fn ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) ∧ (𝑓 ∈ V ∧ 𝑍 ∈ 𝐵) ∧ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅) → ((𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶) ↔ (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍})))
64	57, 59, 60, 62, 63	syl121anc 1376	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶) ↔ (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍})))
65	18, 45, 64	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶) ↔ (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍})))
66	52, 65	mpbid 231	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍}))
67	42, 66	uneq12d 4165	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → ((𝑓 ↾ 𝐶) ∪ (𝑓 ↾ (𝐴 ∖ 𝐶))) = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))
68	41, 67	eqtrd 2773	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))
69	31, 68	jca 513	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍}))))
70	24	ad2antrr 725	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐵 ∈ 𝑊)
71	25	ad2antrr 725	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐴 ∈ 𝑉)
72		elmapi 8843	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐶) → 𝑔:𝐶⟶𝐵)
73	72	ad2antrl 727	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑔:𝐶⟶𝐵)
74		simplr 768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑍 ∈ 𝐵)
75		fconst6g 6781	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝐵 → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶𝐵)
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶𝐵)
77		disjdif 4472	. . . . . . . . 9 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅
78	77	a1i 11	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅)
79		fun2 6755	. . . . . . . 8 ⊢ (((𝑔:𝐶⟶𝐵 ∧ ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶𝐵) ∧ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})):(𝐶 ∪ (𝐴 ∖ 𝐶))⟶𝐵)
80	73, 76, 78, 79	syl21anc 837	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})):(𝐶 ∪ (𝐴 ∖ 𝐶))⟶𝐵)
81		simprr 772	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))
82	81	eqcomd 2739	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) = 𝑓)
83	21	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐶 ⊆ 𝐴)
84	83, 33	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
85	82, 84	feq12d 6706	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})):(𝐶 ∪ (𝐴 ∖ 𝐶))⟶𝐵 ↔ 𝑓:𝐴⟶𝐵))
86	80, 85	mpbid 231	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑓:𝐴⟶𝐵)
87		elmapg 8833	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
88	87	biimpar 479	. . . . . 6 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴⟶𝐵) → 𝑓 ∈ (𝐵 ↑m 𝐴))
89	70, 71, 86, 88	syl21anc 837	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑓 ∈ (𝐵 ↑m 𝐴))
90	71, 74, 86, 47	syl3anc 1372	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑓 supp 𝑍) = (◡𝑓 “ (𝐵 ∖ {𝑍})))
91	81	adantr 482	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))
92	91	fveq1d 6894	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (𝑓‘𝑥) = ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍}))‘𝑥))
93	73	adantr 482	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑔:𝐶⟶𝐵)
94	93	ffnd 6719	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑔 Fn 𝐶)
95		fconstg 6779	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝐵 → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍})
96	95	ad3antlr 730	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍})
97	96	ffnd 6719	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝐴 ∖ 𝐶) × {𝑍}) Fn (𝐴 ∖ 𝐶))
98	77	a1i 11	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅)
99		simpr 486	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑥 ∈ (𝐴 ∖ 𝐶))
100		fvun2 6984	. . . . . . . . 9 ⊢ ((𝑔 Fn 𝐶 ∧ ((𝐴 ∖ 𝐶) × {𝑍}) Fn (𝐴 ∖ 𝐶) ∧ ((𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅ ∧ 𝑥 ∈ (𝐴 ∖ 𝐶))) → ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍}))‘𝑥) = (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥))
101	94, 97, 98, 99, 100	syl112anc 1375	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍}))‘𝑥) = (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥))
102		fvconst 7162	. . . . . . . . 9 ⊢ ((((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥) = 𝑍)
103	96, 99, 102	syl2anc 585	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥) = 𝑍)
104	92, 101, 103	3eqtrd 2777	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (𝑓‘𝑥) = 𝑍)
105	86, 104	suppss 8179	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
106	90, 105	eqsstrrd 4022	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
107	81	reseq1d 5981	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑓 ↾ 𝐶) = ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ↾ 𝐶))
108		res0 5986	. . . . . . . . . 10 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = ∅
109		res0 5986	. . . . . . . . . 10 ⊢ (𝑔 ↾ ∅) = ∅
110	108, 109	eqtr4i 2764	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
111	77	reseq2i 5979	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅)
112	77	reseq2i 5979	. . . . . . . . 9 ⊢ (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (𝑔 ↾ ∅)
113	110, 111, 112	3eqtr4ri 2772	. . . . . . . 8 ⊢ (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶)))
114	113	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))))
115		fresaunres1 6765	. . . . . . 7 ⊢ ((𝑔:𝐶⟶𝐵 ∧ ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶𝐵 ∧ (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶)))) → ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
116	73, 76, 114, 115	syl3anc 1372	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
117	107, 116	eqtr2d 2774	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑔 = (𝑓 ↾ 𝐶))
118	89, 106, 117	jca31 516	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶)))
119	69, 118	impbida 800	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → (((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶)) ↔ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))))
120	16, 119	bitrid 283	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝑓 ∈ 𝑋 ∧ 𝑔 = (𝑓 ↾ 𝐶)) ↔ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))))
121	1, 3, 13, 120	f1od 7658	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  {csn 4629   ↦ cmpt 5232   × cxp 5675  ^◡ccnv 5676   ↾ cres 5679   “ cima 5680   Fn wfn 6539  ⟶wf 6540  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409   supp csupp 8146   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-supp 8147  df-map 8822

This theorem is referenced by:  eulerpartgbij  33402