mapssbi

Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	mapssbi.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	mapssbi.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	mapssbi.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑍 )
	mapssbi.n	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
Assertion	mapssbi	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mapssbi.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		mapssbi.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		mapssbi.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑍 )
4		mapssbi.n	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
5	2	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ 𝑊 )
6		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
7		mapss	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
8	5 6 7	syl2anc	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
9	8	ex	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )
10		simplr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
11		nssrex	⊢ ( ¬ 𝐴 ⊆ 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 )
12	11	bilani	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 )
13		fconst6g	⊢ ( 𝑥 ∈ 𝐴 → ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 )
14	13	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 )
15		elmapg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍 ) → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
16	1 3 15	syl2anc	⊢ ( 𝜑 → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
17	16	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
18	14 17	mpbird	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) )
19	18	3adant3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) )
20	3	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐶 ∈ 𝑍 )
21	2	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐵 ∈ 𝑊 )
22	4	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐶 ≠ ∅ )
23		simpr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
24	20 21 22 23	snelmap	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝑥 ∈ 𝐵 )
25	24	adantlr	⊢ ( ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝑥 ∈ 𝐵 )
26		simplr	⊢ ( ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ¬ 𝑥 ∈ 𝐵 )
27	25 26	pm2.65da	⊢ ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
28	27	3adant2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
29		nelss	⊢ ( ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ∧ ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
30	19 28 29	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
31	30	3exp	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ) )
32	31	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ) )
33	32	rexlimdv	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )
34	12 33	mpd	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
35	34	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
36	10 35	condan	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) → 𝐴 ⊆ 𝐵 )
37	36	ex	⊢ ( 𝜑 → ( ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) → 𝐴 ⊆ 𝐵 ) )
38	9 37	impbid	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )

Metamath Proof Explorer

Theorem mapssbi

Proof