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	biimpi	⊢ ( ¬ 𝐴 ⊆ 𝐵 → ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 )
13	12	adantl	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 )
14		fconst6g	⊢ ( 𝑥 ∈ 𝐴 → ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 )
15	14	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 )
16		elmapg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍 ) → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
17	1 3 16	syl2anc	⊢ ( 𝜑 → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
18	17	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ↔ ( 𝐶 × { 𝑥 } ) : 𝐶 ⟶ 𝐴 ) )
19	15 18	mpbird	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) )
20	19	3adant3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) )
21	3	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐶 ∈ 𝑍 )
22	2	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐵 ∈ 𝑊 )
23	4	adantr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝐶 ≠ ∅ )
24		simpr	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
25	21 22 23 24	snelmap	⊢ ( ( 𝜑 ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝑥 ∈ 𝐵 )
26	25	adantlr	⊢ ( ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → 𝑥 ∈ 𝐵 )
27		simplr	⊢ ( ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) ∧ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ¬ 𝑥 ∈ 𝐵 )
28	26 27	pm2.65da	⊢ ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
29	28	3adant2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) )
30		nelss	⊢ ( ( ( 𝐶 × { 𝑥 } ) ∈ ( 𝐴 ↑_m 𝐶 ) ∧ ¬ ( 𝐶 × { 𝑥 } ) ∈ ( 𝐵 ↑_m 𝐶 ) ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
31	20 29 30	syl2anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
32	31	3exp	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ) )
33	32	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ) )
34	33	rexlimdv	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )
35	13 34	mpd	⊢ ( ( 𝜑 ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
36	35	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) ∧ ¬ 𝐴 ⊆ 𝐵 ) → ¬ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) )
37	10 36	condan	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) → 𝐴 ⊆ 𝐵 )
38	37	ex	⊢ ( 𝜑 → ( ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) → 𝐴 ⊆ 𝐵 ) )
39	9 38	impbid	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑_m 𝐶 ) ⊆ ( 𝐵 ↑_m 𝐶 ) ) )

Metamath Proof Explorer

Theorem mapssbi

Proof