Theorem fnsuppres 8232

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 6682	. . . . . 6 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → dom 𝐹 = (𝐴 ∪ 𝐵))
2	1	rabeqdv 3459	. . . . 5 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
3	2	3ad2ant1 1133	. . . 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 4040	. . 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 4213	. . . . 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 4103	. . . . . 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 4343	. . . . . 6 ⊢ {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} = ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍})
9	8	sseq1i 4037	. . . . 5 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
10	5, 7, 9	3bitr4ri 304	. . . 4 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴)
11		rabss 4095	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴))
12		fvres 6939	. . . . . . . . 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 1200	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑍 ∈ 𝑉)
15		fvconst2g 7239	. . . . . . . . 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 2756	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹‘𝑎) = 𝑍))
18		nne 2950	. . . . . . . 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 1138	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
22		minel 4489	. . . . . . . . 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 308	. . . . . 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 3182	. . . . 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	bitrid 283	. . . 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	bitrid 283	. . 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 279	. 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 6679	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → Fun 𝐹)
32	31	3anim1i 1152	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
33	32	3expb 1120	. . . . 5 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
34		suppval1 8207	. . . . 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 1132	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
37	36	sseq1d 4040	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
38		simp1 1136	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn (𝐴 ∪ 𝐵))
39		ssun2 4202	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
40	39	a1i 11	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
41		fnssres 6703	. . . 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 6809	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
44	43	adantl 481	. . . 4 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45	44	3ad2ant2 1134	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46		eqfnfv 7064	. . 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 311	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   × cxp 5698  dom cdm 5700   ↾ cres 5702  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by:  fnsuppeq0  8233  frlmsslss2  21818  resf1o  32744