Theorem fnsuppres 7322

Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)

Assertion

Ref	Expression
fnsuppres	⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹 ↾ 𝐵) = (𝐵 × {𝑍})))

Proof of Theorem fnsuppres

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fndm 5990	. . . . . 6 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → dom 𝐹 = (𝐴 ∪ 𝐵))
2		rabeq 3192	. . . . . 6 ⊢ (dom 𝐹 = (𝐴 ∪ 𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
3	1, 2	syl 17	. . . . 5 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
4	3	3ad2ant1 1082	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
5	4	sseq1d 3632	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
6		unss 3787	. . . . 5 ⊢ (({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
7		ssrab2 3687	. . . . . 6 ⊢ {𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴
8	7	biantrur 527	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
9		rabun2 3906	. . . . . 6 ⊢ {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} = ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍})
10	9	sseq1i 3629	. . . . 5 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
11	6, 8, 10	3bitr4ri 293	. . . 4 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴)
12		rabss 3679	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴))
13		fvres 6207	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
14	13	adantl 482	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
15		simp2r 1088	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑍 ∈ 𝑉)
16		fvconst2g 6467	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑎 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
17	15, 16	sylan 488	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
18	14, 17	eqeq12d 2637	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹‘𝑎) = 𝑍))
19		nne 2798	. . . . . . . 8 ⊢ (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ (𝐹‘𝑎) = 𝑍)
20	19	a1i 11	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ (𝐹‘𝑎) = 𝑍))
21		id 22	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝐵)
22		simp3 1063	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
23		minel 4033	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑎 ∈ 𝐴)
24	21, 22, 23	syl2anr 495	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ¬ 𝑎 ∈ 𝐴)
25		mtt 354	. . . . . . . 8 ⊢ (¬ 𝑎 ∈ 𝐴 → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴)))
26	24, 25	syl 17	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴)))
27	18, 20, 26	3bitr2rd 297	. . . . . 6 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴) ↔ ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
28	27	ralbidva 2985	. . . . 5 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (∀𝑎 ∈ 𝐵 ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴) ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
29	12, 28	syl5bb 272	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
30	11, 29	syl5bb 272	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
31	5, 30	bitrd 268	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
32		fnfun 5988	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → Fun 𝐹)
33	32	3anim1i 1248	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
34	33	3expb 1266	. . . . 5 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
35		suppval1 7301	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
36	34, 35	syl 17	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
37	36	3adant3 1081	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
38	37	sseq1d 3632	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
39		simp1 1061	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn (𝐴 ∪ 𝐵))
40		ssun2 3777	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
41	40	a1i 11	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
42		fnssres 6004	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝐹 ↾ 𝐵) Fn 𝐵)
43	39, 41, 42	syl2anc 693	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ↾ 𝐵) Fn 𝐵)
44		fnconstg 6093	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
45	44	adantl 482	. . . 4 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
46	45	3ad2ant2 1083	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
47		eqfnfv 6311	. . 3 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐵 × {𝑍}) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
48	43, 46, 47	syl2anc 693	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ↾ 𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
49	31, 38, 48	3bitr4d 300	1 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹 ↾ 𝐵) = (𝐵 × {𝑍})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   × cxp 5112  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   supp csupp 7295

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-supp 7296

This theorem is referenced by:  fnsuppeq0  7323  frlmsslss2  20114  resf1o  29505