Theorem setsfun0

Description: A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun is useful for proofs based on isstruct2 which requires Fun ( F \ { (/) } ) for F to be an extensible structure. (Contributed by AV, 7-Jun-2021)

	Ref	Expression
Assertion	setsfun0	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun ( ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) ∖ { ∅ } ) )

Proof

Step	Hyp	Ref	Expression
1		funres	⊢ ( Fun ( 𝐺 ∖ { ∅ } ) → Fun ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) )
2	1	adantl	⊢ ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) → Fun ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) )
3	2	adantr	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) )
4		funsng	⊢ ( ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) → Fun { ⟨ 𝐼 , 𝐸 ⟩ } )
5	4	adantl	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun { ⟨ 𝐼 , 𝐸 ⟩ } )
6		dmres	⊢ dom ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) = ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) )
7	6	ineq1i	⊢ ( dom ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ( ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } )
8		in32	⊢ ( ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ( ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) )
9		disjdifr	⊢ ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ∅
10	9	ineq1i	⊢ ( ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) ) = ( ∅ ∩ dom ( 𝐺 ∖ { ∅ } ) )
11		0in	⊢ ( ∅ ∩ dom ( 𝐺 ∖ { ∅ } ) ) = ∅
12	8 10 11	3eqtri	⊢ ( ( ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ∩ dom ( 𝐺 ∖ { ∅ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ∅
13	7 12	eqtri	⊢ ( dom ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ∅
14	13	a1i	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( dom ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ∅ )
15		funun	⊢ ( ( ( Fun ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∧ Fun { ⟨ 𝐼 , 𝐸 ⟩ } ) ∧ ( dom ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∩ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) = ∅ ) → Fun ( ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
16	3 5 14 15	syl21anc	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun ( ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
17		difundir	⊢ ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) = ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∖ { ∅ } ) ∪ ( { ⟨ 𝐼 , 𝐸 ⟩ } ∖ { ∅ } ) )
18		resdifcom	⊢ ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∖ { ∅ } ) = ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) )
19	18	a1i	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∖ { ∅ } ) = ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) )
20		elex	⊢ ( 𝐼 ∈ 𝑈 → 𝐼 ∈ V )
21		elex	⊢ ( 𝐸 ∈ 𝑊 → 𝐸 ∈ V )
22	20 21	anim12i	⊢ ( ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) → ( 𝐼 ∈ V ∧ 𝐸 ∈ V ) )
23		opnz	⊢ ( ⟨ 𝐼 , 𝐸 ⟩ ≠ ∅ ↔ ( 𝐼 ∈ V ∧ 𝐸 ∈ V ) )
24	22 23	sylibr	⊢ ( ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) → ⟨ 𝐼 , 𝐸 ⟩ ≠ ∅ )
25	24	adantl	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ⟨ 𝐼 , 𝐸 ⟩ ≠ ∅ )
26		disjsn2	⊢ ( ⟨ 𝐼 , 𝐸 ⟩ ≠ ∅ → ( { ⟨ 𝐼 , 𝐸 ⟩ } ∩ { ∅ } ) = ∅ )
27		disjdif2	⊢ ( ( { ⟨ 𝐼 , 𝐸 ⟩ } ∩ { ∅ } ) = ∅ → ( { ⟨ 𝐼 , 𝐸 ⟩ } ∖ { ∅ } ) = { ⟨ 𝐼 , 𝐸 ⟩ } )
28	25 26 27	3syl	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( { ⟨ 𝐼 , 𝐸 ⟩ } ∖ { ∅ } ) = { ⟨ 𝐼 , 𝐸 ⟩ } )
29	19 28	uneq12d	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∖ { ∅ } ) ∪ ( { ⟨ 𝐼 , 𝐸 ⟩ } ∖ { ∅ } ) ) = ( ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
30	17 29	eqtrid	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) = ( ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
31	30	funeqd	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fun ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) ↔ Fun ( ( ( 𝐺 ∖ { ∅ } ) ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ) )
32	16 31	mpbird	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) )
33		opex	⊢ ⟨ 𝐼 , 𝐸 ⟩ ∈ V
34	33	a1i	⊢ ( Fun ( 𝐺 ∖ { ∅ } ) → ⟨ 𝐼 , 𝐸 ⟩ ∈ V )
35		setsvalg	⊢ ( ( 𝐺 ∈ 𝑉 ∧ ⟨ 𝐼 , 𝐸 ⟩ ∈ V ) → ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) = ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
36	34 35	sylan2	⊢ ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) → ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) = ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) )
37	36	difeq1d	⊢ ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) → ( ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) ∖ { ∅ } ) = ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) )
38	37	funeqd	⊢ ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) → ( Fun ( ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) ∖ { ∅ } ) ↔ Fun ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) ) )
39	38	adantr	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fun ( ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) ∖ { ∅ } ) ↔ Fun ( ( ( 𝐺 ↾ ( V ∖ dom { ⟨ 𝐼 , 𝐸 ⟩ } ) ) ∪ { ⟨ 𝐼 , 𝐸 ⟩ } ) ∖ { ∅ } ) ) )
40	32 39	mpbird	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ Fun ( 𝐺 ∖ { ∅ } ) ) ∧ ( 𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊 ) ) → Fun ( ( 𝐺 sSet ⟨ 𝐼 , 𝐸 ⟩ ) ∖ { ∅ } ) )