mapss2 - Mathbox for Glauco Siliprandi

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Theorem mapss2 45112

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 8947	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
5	2, 3, 4	syl2anc 583	. . 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 4376	. . . . . 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 2741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦))
12		eqidd 2741	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
14		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ V)
16	11, 12, 13, 15	fvmptd 7036	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) = 𝑦)
17	16	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
18	17	ad4ant13 750	. . . . . . . . 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 7149	. . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶))
29	19, 28	sseldd 4009	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑_m 𝐶))
30		elmapi 8907	. . . . . . . . . . . 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 714	. . . . . . . . . 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 7119	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) ∈ 𝐵)
35	18, 34	eqeltrd 2844	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
36	35	ralrimiva 3152	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
37		dfss3 3997	. . . . . . 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 1932	. . . 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 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 Vcvv 3488 ⊆ wss 3976 ∅c0 4352 ↦ cmpt 5249 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886

This theorem is referenced by: ovnovollem1 46577 ovnovollem2 46578

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