mapss2 - Mathbox for Glauco Siliprandi

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Theorem mapss2 45209

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

**Hypotheses**
Ref	Expression
mapss2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapss2.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
mapss2.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)
mapss2.n	⊢ (𝜑 → 𝐶 ≠ ∅)

**Assertion**
Ref	Expression
mapss2	⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Proof of Theorem mapss2 Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapss2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8908	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
5	2, 3, 4	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
6	5	ex 412	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
7		mapss2.n	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ ∅)
8		n0 4333	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
11		eqidd 2737	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦))
12		eqidd 2737	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
14		vex 3468	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ V)
16	11, 12, 13, 15	fvmptd 6998	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) = 𝑦)
17	16	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
18	17	ad4ant13 751	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
19		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
20		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑦 ∈ 𝐴)
21	20	fmpttd 7110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴)
22		mapss2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ 𝑉)
24		mapss2.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
25	24	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐶 ∈ 𝑍)
26	23, 25	elmapd 8859	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶) ↔ (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴))
27	21, 26	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶))
28	27	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶))
29	19, 28	sseldd 3964	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑_m 𝐶))
30		elmapi 8868	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑_m 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
32	31	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
33		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐶)
34	32, 33	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) ∈ 𝐵)
35	18, 34	eqeltrd 2835	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
36	35	ralrimiva 3133	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
37		dfss3 3952	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
38	36, 37	sylibr 234	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐵)
39	38	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → (𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
40	39	exlimdv 1933	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → (∃𝑥 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
41	10, 40	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → 𝐴 ⊆ 𝐵)
42	41	ex 412	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → 𝐴 ⊆ 𝐵))
43	6, 42	impbid 212	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 Vcvv 3464 ⊆ wss 3931 ∅c0 4313 ↦ cmpt 5206 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-map 8847

This theorem is referenced by: ovnovollem1 46665 ovnovollem2 46666

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