Theorem mapss2 40202

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 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8173	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
5	2, 3, 4	syl2anc 579	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
6	5	ex 403	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
7		mapss2.n	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ ∅)
8		n0 4162	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
9	7, 8	sylib 210	. . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
10	9	adantr 474	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
11		eqidd 2826	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦))
12		eqidd 2826	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
14		vex 3417	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ V)
16	11, 12, 13, 15	fvmptd 6539	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) = 𝑦)
17	16	eqcomd 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
18	17	ad4ant13 757	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
19		simplr 785	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
20		simplr 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑦 ∈ 𝐴)
21	20	fmpttd 6639	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴)
22		mapss2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23	22	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ 𝑉)
24		mapss2.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
25	24	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐶 ∈ 𝑍)
26	23, 25	elmapd 8141	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶) ↔ (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴))
27	21, 26	mpbird 249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶))
28	27	adantlr 706	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶))
29	19, 28	sseldd 3828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑𝑚 𝐶))
30		elmapi 8149	. . . . . . . . . . . 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 706	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
33		simplr 785	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐶)
34	32, 33	ffvelrnd 6614	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) ∈ 𝐵)
35	18, 34	eqeltrd 2906	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
36	35	ralrimiva 3175	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
37		dfss3 3816	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
38	36, 37	sylibr 226	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐵)
39	38	ex 403	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → (𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
40	39	exlimdv 2032	. . . 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 403	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶) → 𝐴 ⊆ 𝐵))
43	6, 42	impbid 204	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∅c0 4146   ↦ cmpt 4954  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910   ↑_𝑚 cmap 8127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129

This theorem is referenced by:  ovnovollem1  41662  ovnovollem2  41663