Theorem resf1o 31061

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 5936	. . 3 ⊢ (𝑓 ∈ 𝑋 → (𝑓 ↾ 𝐶) ∈ V)
3	2	adantl 482	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ 𝐶) ∈ V)
4		simpr 485	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → 𝑔 ∈ (𝐵 ↑m 𝐶))
5		difexg 5255	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐶) ∈ V)
6	5	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∖ 𝐶) ∈ V)
7		snex 5358	. . . . . 6 ⊢ {𝑍} ∈ V
8		xpexg 7594	. . . . . 6 ⊢ (((𝐴 ∖ 𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
9	6, 7, 8	sylancl 586	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
10	9	adantr 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V)
11		unexg 7593	. . . 4 ⊢ ((𝑔 ∈ (𝐵 ↑m 𝐶) ∧ ((𝐴 ∖ 𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
12	4, 10, 11	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑔 ∈ (𝐵 ↑m 𝐶)) → (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})) ∈ V)
13	12	adantlr 712	. 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 3421	. . . 4 ⊢ (𝑓 ∈ 𝑋 ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
16	15	anbi1i 624	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 = (𝑓 ↾ 𝐶)) ↔ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶)))
17		simprr 770	. . . . . 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 776	. . . . . . . . 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 8620	. . . . . . . . 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 1137	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
22	21	ad2antrr 723	. . . . . . . 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 6639	. . . . . . 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 1136	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ 𝑊)
25		simp1 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ 𝑉)
26	25, 21	ssexd 5252	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V)
27		elmapg 8611	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ V) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
28	24, 26, 27	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) → ((𝑓 ↾ 𝐶) ∈ (𝐵 ↑m 𝐶) ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐵))
29	28	ad2antrr 723	. . . . . . 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 256	. . . . . 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 2841	. . . . 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 4421	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
33	32	biimpi 215	. . . . . . . . . 10 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴)
34	33	reseq2d 5890	. . . . . . . . 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 6598	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
37		fnresdm 6549	. . . . . . . . 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 2781	. . . . . . 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 5904	. . . . . . 7 ⊢ (𝑓 ↾ (𝐶 ∪ (𝐴 ∖ 𝐶))) = ((𝑓 ↾ 𝐶) ∪ (𝑓 ↾ (𝐴 ∖ 𝐶)))
41	39, 40	eqtrdi 2796	. . . . . 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 2746	. . . . . . 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 777	. . . . . . . . 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 723	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝐴 ∈ 𝑉)
45		simplr 766	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓 ↾ 𝐶))) → 𝑍 ∈ 𝐵)
46		eqid 2740	. . . . . . . . . . 11 ⊢ (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
47	46	ffs2 31059	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵 ∧ 𝑓:𝐴⟶𝐵) → (𝑓 supp 𝑍) = (◡𝑓 “ (𝐵 ∖ {𝑍})))
48	44, 45, 20, 47	syl3anc 1370	. . . . . . . . 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 4155	. . . . . . . . . . 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 3973	. . . . . . . 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 483	. . . . . . . . . . . 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 4418	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐴 ∖ 𝐶)) = 𝐴
56	55	fneq2i 6529	. . . . . . . . . . 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 3435	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑓 ∈ V)
60		simpr 485	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
61		inindif 30859	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅)
63		fnsuppres 7998	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑍 ∈ 𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴 ∩ 𝐶) ↔ (𝑓 ↾ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) × {𝑍})))
65	18, 45, 64	syl2anc 584	. . . . . . . 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 4103	. . . . . 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 2780	. . . . 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 512	. . . 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 723	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐵 ∈ 𝑊)
71	25	ad2antrr 723	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝐴 ∈ 𝑉)
72		elmapi 8620	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐶) → 𝑔:𝐶⟶𝐵)
73	72	ad2antrl 725	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑔:𝐶⟶𝐵)
74		simplr 766	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) ∧ (𝑔 ∈ (𝐵 ↑m 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴 ∖ 𝐶) × {𝑍})))) → 𝑍 ∈ 𝐵)
75		fconst6g 6661	. . . . . . . . 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 4411	. . . . . . . . 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 6635	. . . . . . . 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 835	. . . . . . 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 770	. . . . . . . . 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 2746	. . . . . . . 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 723	. . . . . . . . 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 6586	. . . . . . 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 8611	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
88	87	biimpar 478	. . . . . 6 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ 𝑓:𝐴⟶𝐵) → 𝑓 ∈ (𝐵 ↑m 𝐴))
89	70, 71, 86, 88	syl21anc 835	. . . . 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 1370	. . . . . 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 481	. . . . . . . . 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 6773	. . . . . . . 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 481	. . . . . . . . . 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 6599	. . . . . . . . 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 6659	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝐵 → ((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍})
96	95	ad3antlr 728	. . . . . . . . . 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 6599	. . . . . . . . 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 485	. . . . . . . . 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 6857	. . . . . . . . 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 1373	. . . . . . . 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 7033	. . . . . . . . 9 ⊢ ((((𝐴 ∖ 𝐶) × {𝑍}):(𝐴 ∖ 𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (((𝐴 ∖ 𝐶) × {𝑍})‘𝑥) = 𝑍)
103	96, 99, 102	syl2anc 584	. . . . . . . 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 2784	. . . . . . 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 8001	. . . . . 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 3965	. . . . 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 5889	. . . . . 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 5894	. . . . . . . . . 10 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = ∅
109		res0 5894	. . . . . . . . . 10 ⊢ (𝑔 ↾ ∅) = ∅
110	108, 109	eqtr4i 2771	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
111	77	reseq2i 5887	. . . . . . . . 9 ⊢ (((𝐴 ∖ 𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (((𝐴 ∖ 𝐶) × {𝑍}) ↾ ∅)
112	77	reseq2i 5887	. . . . . . . . 9 ⊢ (𝑔 ↾ (𝐶 ∩ (𝐴 ∖ 𝐶))) = (𝑔 ↾ ∅)
113	110, 111, 112	3eqtr4ri 2779	. . . . . . . 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 6645	. . . . . . 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 1370	. . . . . 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 2781	. . . . 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 515	. . . 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 798	. . 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 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 7515	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  {crab 3070  Vcvv 3431   ∖ cdif 3889   ∪ cun 3890   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  {csn 4567   ↦ cmpt 5162   × cxp 5588  ^◡ccnv 5589   ↾ cres 5592   “ cima 5593   Fn wfn 6427  ⟶wf 6428  –1-1-onto→wf1o 6431  ‘cfv 6432  (class class class)co 7271   supp csupp 7968   ↑_m cmap 8598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-supp 7969  df-map 8600

This theorem is referenced by:  eulerpartgbij  32335