Theorem fnsuppres 7978

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 6520	. . . . . 6 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → dom 𝐹 = (𝐴 ∪ 𝐵))
2	1	rabeqdv 3409	. . . . 5 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
3	2	3ad2ant1 1131	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
4	3	sseq1d 3948	. . 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))
5		unss 4114	. . . . 5 ⊢ (({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
6		ssrab2 4009	. . . . . 6 ⊢ {𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴
7	6	biantrur 530	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
8		rabun2 4244	. . . . . 6 ⊢ {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} = ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍})
9	8	sseq1i 3945	. . . . 5 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
10	5, 7, 9	3bitr4ri 303	. . . 4 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴)
11		rabss 4001	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴))
12		fvres 6775	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
13	12	adantl 481	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
14		simp2r 1198	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑍 ∈ 𝑉)
15		fvconst2g 7059	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑎 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
16	14, 15	sylan 579	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
17	13, 16	eqeq12d 2754	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹‘𝑎) = 𝑍))
18		nne 2946	. . . . . . . 8 ⊢ (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ (𝐹‘𝑎) = 𝑍)
19	18	a1i 11	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ (𝐹‘𝑎) = 𝑍))
20		id 22	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝐵)
21		simp3 1136	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
22		minel 4396	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑎 ∈ 𝐴)
23	20, 21, 22	syl2anr 596	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ¬ 𝑎 ∈ 𝐴)
24		mtt 364	. . . . . . . 8 ⊢ (¬ 𝑎 ∈ 𝐴 → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴)))
25	23, 24	syl 17	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (¬ (𝐹‘𝑎) ≠ 𝑍 ↔ ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴)))
26	17, 19, 25	3bitr2rd 307	. . . . . 6 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴) ↔ ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
27	26	ralbidva 3119	. . . . 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)))
28	11, 27	syl5bb 282	. . . 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)))
29	10, 28	syl5bb 282	. . 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)))
30	4, 29	bitrd 278	. 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)))
31		fnfun 6517	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → Fun 𝐹)
32	31	3anim1i 1150	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
33	32	3expb 1118	. . . . 5 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
34		suppval1 7954	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
35	33, 34	syl 17	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
36	35	3adant3 1130	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
37	36	sseq1d 3948	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
38		simp1 1134	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn (𝐴 ∪ 𝐵))
39		ssun2 4103	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
40	39	a1i 11	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
41		fnssres 6539	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝐹 ↾ 𝐵) Fn 𝐵)
42	38, 40, 41	syl2anc 583	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ↾ 𝐵) Fn 𝐵)
43		fnconstg 6646	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
44	43	adantl 481	. . . 4 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45	44	3ad2ant2 1132	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46		eqfnfv 6891	. . 3 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐵 × {𝑍}) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
47	42, 45, 46	syl2anc 583	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ↾ 𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
48	30, 37, 47	3bitr4d 310	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558   × cxp 5578  dom cdm 5580   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   supp csupp 7948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-supp 7949

This theorem is referenced by:  fnsuppeq0  7979  frlmsslss2  20892  resf1o  30967