mapssbi - Mathbox for Glauco Siliprandi

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Theorem mapssbi 41496

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 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8453	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
5	2, 3, 4	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
6	5	ex 415	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
7		simplr 767	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
8		nssrex 41371	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9	8	biimpi 218	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
10	9	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
11		fconst6g 6568	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 × {𝑥}):𝐶⟶𝐴)
12	11	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}):𝐶⟶𝐴)
13		mapssbi.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14		mapssbi.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
15		elmapg 8419	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
16	13, 14, 15	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
17	16	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
18	12, 17	mpbird 259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
19	18	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
20	14	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ∈ 𝑍)
21	1	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐵 ∈ 𝑊)
22		mapssbi.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ ∅)
23	22	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ≠ ∅)
24		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
25	20, 21, 23, 24	snelmap 41366	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
26	25	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
27		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ 𝑥 ∈ 𝐵)
28	26, 27	pm2.65da 815	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
29	28	3adant2 1127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
30		nelss 4030	. . . . . . . . . 10 ⊢ (((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
31	19, 29, 30	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
32	31	3exp 1115	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
33	32	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
34	33	rexlimdv 3283	. . . . . 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 713	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
37	7, 36	condan 816	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → 𝐴 ⊆ 𝐵)
38	37	ex 415	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → 𝐴 ⊆ 𝐵))
39	6, 38	impbid 214	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 {csn 4567 × cxp 5553 ⟶wf 6351 (class class class)co 7156 ↑_m cmap 8406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408

This theorem is referenced by: (None)

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