Theorem mapssbi 39405

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

Hypotheses

Ref	Expression
mapssbi.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapssbi.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
mapssbi.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)
mapssbi.n	⊢ (𝜑 → 𝐶 ≠ ∅)

Assertion

Ref	Expression
mapssbi	⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))

Proof of Theorem mapssbi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapssbi.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 7900	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
5	2, 3, 4	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
6	5	ex 450	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
7		simplr 792	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
8		nssrex 39260	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9	8	biimpi 206	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
11		fconst6g 6094	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 × {𝑥}):𝐶⟶𝐴)
12	11	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}):𝐶⟶𝐴)
13		mapssbi.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14		mapssbi.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
15		elmapg 7870	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
16	13, 14, 15	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
18	12, 17	mpbird 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶))
19	18	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶))
20	14	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → 𝐶 ∈ 𝑍)
21	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → 𝐵 ∈ 𝑊)
22		mapssbi.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ ∅)
23	22	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → 𝐶 ≠ ∅)
24		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶))
25	20, 21, 23, 24	snelmap 39254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → 𝑥 ∈ 𝐵)
26	25	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → 𝑥 ∈ 𝐵)
27		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → ¬ 𝑥 ∈ 𝐵)
28	26, 27	pm2.65da 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶))
29	28	3adant2 1080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶))
30		nelss 3664	. . . . . . . . . 10 ⊢ (((𝐶 × {𝑥}) ∈ (𝐴 ↑𝑚 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑𝑚 𝐶)) → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
31	19, 29, 30	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
32	31	3exp 1264	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))))
33	32	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))))
34	33	rexlimdv 3030	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
35	10, 34	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
36	35	adantlr 751	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
37	7, 36	condan 835	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → 𝐴 ⊆ 𝐵)
38	37	ex 450	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶) → 𝐴 ⊆ 𝐵))
39	6, 38	impbid 202	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  {csn 4177   × cxp 5112  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)