Theorem mapss2 39397

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 477	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 7900	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
5	2, 3, 4	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
6	5	ex 450	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))
7		mapss2.n	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ ∅)
8		n0 3931	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
9	7, 8	sylib 208	. . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
11		eqidd 2623	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦))
12		eqidd 2623	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
14		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ V)
16	11, 12, 13, 15	fvmptd 6288	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) = 𝑦)
17	16	eqcomd 2628	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
18	17	ad4ant13 1292	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 = ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥))
19		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶))
20		simplr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑦 ∈ 𝐴)
21		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝐶 ↦ 𝑦) = (𝑤 ∈ 𝐶 ↦ 𝑦)
22	20, 21	fmptd 6385	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴)
23		mapss2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ 𝑉)
25		mapss2.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
26	25	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐶 ∈ 𝑍)
27	24, 26	elmapd 7871	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶) ↔ (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐴))
28	22, 27	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶))
29	28	adantlr 751	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐴 ↑𝑚 𝐶))
30	19, 29	sseldd 3604	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑𝑚 𝐶))
31		elmapi 7879	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐶 ↦ 𝑦) ∈ (𝐵 ↑𝑚 𝐶) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
33	32	adantlr 751	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → (𝑤 ∈ 𝐶 ↦ 𝑦):𝐶⟶𝐵)
34		simplr 792	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐶)
35	33, 34	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝐶 ↦ 𝑦)‘𝑥) ∈ 𝐵)
36	18, 35	eqeltrd 2701	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
37	36	ralrimiva 2966	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
38		dfss3 3592	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ 𝐵)
39	37, 38	sylibr 224	. . . . . 6 ⊢ (((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐵)
40	39	ex 450	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → (𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
41	40	exlimdv 1861	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → (∃𝑥 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵))
42	10, 41	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) → 𝐴 ⊆ 𝐵)
43	42	ex 450	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶) → 𝐴 ⊆ 𝐵))
44	6, 43	impbid 202	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  ovnovollem1  40870  ovnovollem2  40871