Theorem resf1o 30071

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 5692	. . 3 ⊢ (𝑓 ∈ 𝑋 → (𝑓 ↾ 𝐶) ∈ V)
3	2	adantl 475	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ 𝐶) ∈ V)
4		simpr 479	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑𝑚 𝐶)) → 𝑔 ∈ (𝐵 ↑𝑚 𝐶))
5		difexg 5045	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐶) ∈ V)
6	5	3ad2ant1 1124	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∖ 𝐶) ∈ V)
7		snex 5140	. . . . . 6 ⊢ {𝑍} ∈ V
8		xpexg 7237	. . . . . 6 ⊢ (((𝐴 ∖ 𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
9	6, 7, 8	sylancl 580	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
10	9	adantr 474	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑𝑚 𝐶)) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
11		unexg 7236	. . . 4 ⊢ ((𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
12	4, 10, 11	syl2anc 579	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑𝑚 𝐶)) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
13	12	adantlr 705	. 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	rabeq2i 3394	. . . 4 ⊢ (𝑓 ∈ 𝑋 ↔ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
16	15	anbi1i 617	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 = (𝑓 ↾ 𝐶)) ↔ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶)))
17		simprr 763	. . . . . 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 769	. . . . . . . . 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 8162	. . . . . . . . 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 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
22	21	ad2antrr 716	. . . . . . . 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 6321	. . . . . . 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 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ 𝑊)
25		simp1 1127	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ 𝑉)
26	25, 21	ssexd 5042	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V)
27		elmapg 8153	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ V) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑𝑚 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
28	24, 26, 27	syl2anc 579	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑𝑚 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
29	28	ad2antrr 716	. . . . . . 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 249	. . . . . 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 2859	. . . . 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 4273	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
33	32	biimpi 208	. . . . . . . . . 10 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
34	33	reseq2d 5642	. . . . . . . . 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 6291	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
37		fnresdm 6246	. . . . . . . . 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 2815	. . . . . . 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 5660	. . . . . . 7 ⊢ (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))) = ((𝑓 ↾ 𝐶) ∪ (𝑓 ↾ (𝐴 ∖ 𝐶)))
41	39, 40	syl6eq 2830	. . . . . 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 2784	. . . . . . 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 770	. . . . . . . . 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 716	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝐴 ∈ 𝑉)
45		simplr 759	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑍 ∈ 𝐵)
46		eqid 2778	. . . . . . . . . . 11 ⊢ (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
47	46	ffs2 30069	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵 ∧ 𝑓:𝐴⟶𝐵) → (𝑓 supp 𝑍) = (◡𝑓 “ (𝐵 ∖ {𝑍})))
48	44, 45, 20, 47	syl3anc 1439	. . . . . . . . 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 4040	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
50	49	biimpi 208	. . . . . . . . . 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 3867	. . . . . . . 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 476	. . . . . . . . . . . 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 4270	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
56	55	fneq2i 6231	. . . . . . . . . . 11 ⊢ (𝑓 Fn ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) ↔ 𝑓 Fn 𝐴)
57	54, 56	sylibr 226	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 Fn ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)))
58		vex 3401	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 ∈ V)
60		simpr 479	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
61		inindif 29915	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
63		fnsuppres 7604	. . . . . . . . . 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 1443	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶) ↔ (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍})))
65	18, 45, 64	syl2anc 579	. . . . . . . 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 224	. . . . . . 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 3991	. . . . . 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 2814	. . . . 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 507	. . . 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 716	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐵 ∈ 𝑊)
71	25	ad2antrr 716	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐴 ∈ 𝑉)
72		elmapi 8162	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) → 𝑔:𝐶⟶𝐵)
73	72	ad2antrl 718	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑔:𝐶⟶𝐵)
74		simplr 759	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑍 ∈ 𝐵)
75		fconst6g 6344	. . . . . . . . 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 4264	. . . . . . . . 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 6317	. . . . . . . 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 828	. . . . . . 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 763	. . . . . . . . 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 2784	. . . . . . . 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 716	. . . . . . . . 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 6279	. . . . . . 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 224	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑓:𝐴⟶𝐵)
87		elmapg 8153	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝐵))
88	87	biimpar 471	. . . . . 6 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴⟶𝐵) → 𝑓 ∈ (𝐵 ↑𝑚 𝐴))
89	70, 71, 86, 88	syl21anc 828	. . . . 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 1439	. . . . . 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 474	. . . . . . . . 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 6448	. . . . . . . 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 474	. . . . . . . . . 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 6292	. . . . . . . . 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 6342	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝐵 → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍})
96	95	ad3antlr 721	. . . . . . . . . 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 6292	. . . . . . . . 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 479	. . . . . . . . 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 6530	. . . . . . . . 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 1442	. . . . . . . 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 6697	. . . . . . . . 9 ⊢ ((((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥) = 𝑍)
103	96, 99, 102	syl2anc 579	. . . . . . . 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 2818	. . . . . . 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 7607	. . . . . 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	eqsstr3d 3859	. . . . 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 5641	. . . . . 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 5646	. . . . . . . . . 10 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = ∅
109		res0 5646	. . . . . . . . . 10 ⊢ (𝑔 ↾ ∅) = ∅
110	108, 109	eqtr4i 2805	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
111	77	reseq2i 5639	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅)
112	77	reseq2i 5639	. . . . . . . . 9 ⊢ (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (𝑔 ↾ ∅)
113	110, 111, 112	3eqtr4ri 2813	. . . . . . . 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 6327	. . . . . . 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 1439	. . . . . 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 2815	. . . . 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 510	. . . 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 791	. . 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	syl5bb 275	. 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 7162	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑𝑚 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {crab 3094  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398   ↦ cmpt 4965   × cxp 5353  ^◡ccnv 5354   ↾ cres 5357   “ cima 5358   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   supp csupp 7576   ↑_𝑚 cmap 8140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-supp 7577  df-map 8142

This theorem is referenced by:  eulerpartgbij  31032