Theorem difmapsn 39404

Description: Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
difmapsn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
difmapsn.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
difmapsn.v	⊢ (𝜑 → 𝐶 ∈ 𝑍)

Assertion

Ref	Expression
difmapsn	⊢ (𝜑 → ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) = ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))

Proof of Theorem difmapsn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) → 𝑓 ∈ (𝐴 ↑𝑚 {𝐶}))
2	1	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → 𝑓 ∈ (𝐴 ↑𝑚 {𝐶}))
3		elmapi 7879	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐶}) → 𝑓:{𝐶}⟶𝐴)
4	3	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ↑𝑚 {𝐶})) → 𝑓:{𝐶}⟶𝐴)
5		difmapsn.v	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
6		fsn2g 6405	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝑍 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓‘𝐶) ∈ 𝐴 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓‘𝐶) ∈ 𝐴 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
8	7	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ↑𝑚 {𝐶})) → (𝑓:{𝐶}⟶𝐴 ↔ ((𝑓‘𝐶) ∈ 𝐴 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
9	4, 8	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ↑𝑚 {𝐶})) → ((𝑓‘𝐶) ∈ 𝐴 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩}))
10	9	simpld 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ↑𝑚 {𝐶})) → (𝑓‘𝐶) ∈ 𝐴)
11	2, 10	syldan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → (𝑓‘𝐶) ∈ 𝐴)
12		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → (𝑓‘𝐶) ∈ 𝐵)
13	9	simprd 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ↑𝑚 {𝐶})) → 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})
14	2, 13	syldan 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})
15	14	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})
16	12, 15	jca 554	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → ((𝑓‘𝐶) ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩}))
17		fsn2g 6405	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝑍 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓‘𝐶) ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
18	5, 17	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓‘𝐶) ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
19	18	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → (𝑓:{𝐶}⟶𝐵 ↔ ((𝑓‘𝐶) ∈ 𝐵 ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
20	16, 19	mpbird 247	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → 𝑓:{𝐶}⟶𝐵)
21		difmapsn.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
22	21	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → 𝐵 ∈ 𝑊)
23		snex 4908	. . . . . . . . . . . 12 ⊢ {𝐶} ∈ V
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → {𝐶} ∈ V)
25	22, 24	elmapd 7871	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐶}) ↔ 𝑓:{𝐶}⟶𝐵))
26	20, 25	mpbird 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
27		eldifn 3733	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) → ¬ 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
28	27	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) ∧ (𝑓‘𝐶) ∈ 𝐵) → ¬ 𝑓 ∈ (𝐵 ↑𝑚 {𝐶}))
29	26, 28	pm2.65da 600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → ¬ (𝑓‘𝐶) ∈ 𝐵)
30	11, 29	eldifd 3585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → (𝑓‘𝐶) ∈ (𝐴 ∖ 𝐵))
31	30, 14	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → ((𝑓‘𝐶) ∈ (𝐴 ∖ 𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩}))
32		fsn2g 6405	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → (𝑓:{𝐶}⟶(𝐴 ∖ 𝐵) ↔ ((𝑓‘𝐶) ∈ (𝐴 ∖ 𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
33	5, 32	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑓:{𝐶}⟶(𝐴 ∖ 𝐵) ↔ ((𝑓‘𝐶) ∈ (𝐴 ∖ 𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
34	33	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → (𝑓:{𝐶}⟶(𝐴 ∖ 𝐵) ↔ ((𝑓‘𝐶) ∈ (𝐴 ∖ 𝐵) ∧ 𝑓 = {⟨𝐶, (𝑓‘𝐶)⟩})))
35	31, 34	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → 𝑓:{𝐶}⟶(𝐴 ∖ 𝐵))
36		difmapsn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
37		difssd 3738	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
38	36, 37	ssexd 4805	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
39	23	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐶} ∈ V)
40	38, 39	elmapd 7871	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴 ∖ 𝐵)))
41	40	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → (𝑓 ∈ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}) ↔ 𝑓:{𝐶}⟶(𝐴 ∖ 𝐵)))
42	35, 41	mpbird 247	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))) → 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))
43	42	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))𝑓 ∈ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))
44		dfss3 3592	. . 3 ⊢ (((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) ⊆ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}) ↔ ∀𝑓 ∈ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶}))𝑓 ∈ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))
45	43, 44	sylibr 224	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) ⊆ ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))
46	5	snn0d 39258	. . 3 ⊢ (𝜑 → {𝐶} ≠ ∅)
47	36, 21, 39, 46	difmap 39399	. 2 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}) ⊆ ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})))
48	45, 47	eqssd 3620	1 ⊢ (𝜑 → ((𝐴 ↑𝑚 {𝐶}) ∖ (𝐵 ↑𝑚 {𝐶})) = ((𝐴 ∖ 𝐵) ↑𝑚 {𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ⟶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-reu 2919  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-f1 5893  df-fo 5894  df-f1o 5895  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:  vonvolmbllem  40874  vonvolmbl  40875