Theorem fnsuppres 8138

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 6595	. . . . . 6 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → dom 𝐹 = (𝐴 ∪ 𝐵))
2	1	rabeqdv 3407	. . . . 5 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍})
3	2	3ad2ant1 1139	. . . 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 3953	. . 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 4126	. . . . 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 4018	. . . . . 6 ⊢ {𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴
7	6	biantrur 535	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
8		rabun2 4259	. . . . . 6 ⊢ {𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} = ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍})
9	8	sseq1i 3950	. . . . 5 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎 ∈ 𝐴 ∣ (𝐹‘𝑎) ≠ 𝑍} ∪ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍}) ⊆ 𝐴)
10	5, 7, 9	3bitr4ri 305	. . . 4 ⊢ ({𝑎 ∈ (𝐴 ∪ 𝐵) ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴)
11		rabss 4008	. . . . 5 ⊢ ({𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎 ∈ 𝐵 ((𝐹‘𝑎) ≠ 𝑍 → 𝑎 ∈ 𝐴))
12		fvres 6853	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
13	12	adantl 482	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑎) = (𝐹‘𝑎))
14		simp2r 1207	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑍 ∈ 𝑉)
15		fvconst2g 7153	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑎 ∈ 𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
16	14, 15	sylan 586	. . . . . . . 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 2939	. . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
22		minel 4401	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑎 ∈ 𝐴)
23	20, 21, 22	syl2anr 603	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐵) → ¬ 𝑎 ∈ 𝐴)
24		mtt 365	. . . . . . . 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 309	. . . . . 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 3161	. . . . 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 284	. . . 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 284	. . 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 280	. 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 6592	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 ∪ 𝐵) → Fun 𝐹)
32	31	3anim1i 1158	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
33	32	3expb 1126	. . . . 5 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) → (Fun 𝐹 ∧ 𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))
34		suppval1 8113	. . . . 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 1138	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍})
37	36	sseq1d 3953	. 2 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹‘𝑎) ≠ 𝑍} ⊆ 𝐴))
38		simp1 1142	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn (𝐴 ∪ 𝐵))
39		ssun2 4115	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
40	39	a1i 11	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
41		fnssres 6615	. . . 4 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝐹 ↾ 𝐵) Fn 𝐵)
42	38, 40, 41	syl2anc 590	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ↾ 𝐵) Fn 𝐵)
43		fnconstg 6722	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
44	43	adantl 482	. . . 4 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45	44	3ad2ant2 1140	. . 3 ⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46		eqfnfv 6978	. . 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 590	. 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 312	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  {crab 3392   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562   × cxp 5623  dom cdm 5625   ↾ cres 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363   supp csupp 8107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-supp 8108

This theorem is referenced by:  fnsuppeq0  8139  frlmsslss2  21757  resf1o  32829