mapss2 - Mathbox for Glauco Siliprandi

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Theorem mapss2 42745

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 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8677	. . . 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 413	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
7		mapss2.n	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ ∅)
8		n0 4280	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
9	7, 8	sylib 217	. . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
11		eqidd 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦))
12		eqidd 2739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
14		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ V)
16	11, 12, 13, 15	fvmptd 6882	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) = 𝑦)
17	16	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
18	17	ad4ant13 748	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
19		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
20		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑦 ∈ 𝐴)
21	20	fmpttd 6989	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴)
22		mapss2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23	22	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ 𝑉)
24		mapss2.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
25	24	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐶 ∈ 𝑍)
26	23, 25	elmapd 8629	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶) ↔ (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴))
27	21, 26	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶))
28	27	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑_m 𝐶))
29	19, 28	sseldd 3922	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑_m 𝐶))
30		elmapi 8637	. . . . . . . . . . . 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 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
33		simplr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐶)
34	32, 33	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) ∈ 𝐵)
35	18, 34	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
36	35	ralrimiva 3103	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
37		dfss3 3909	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
38	36, 37	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐵)
39	38	ex 413	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → (𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
40	39	exlimdv 1936	. . . 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 413	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → 𝐴 ⊆ 𝐵))
43	6, 42	impbid 211	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617

This theorem is referenced by: ovnovollem1 44194 ovnovollem2 44195

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