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Theorem fnsuppres 7367
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 (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹𝐵) = (𝐵 × {𝑍})))

Proof of Theorem fnsuppres
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fndm 6028 . . . . . 6 (𝐹 Fn (𝐴𝐵) → dom 𝐹 = (𝐴𝐵))
2 rabeq 3223 . . . . . 6 (dom 𝐹 = (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
31, 2syl 17 . . . . 5 (𝐹 Fn (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
433ad2ant1 1102 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
54sseq1d 3665 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
6 unss 3820 . . . . 5 (({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
7 ssrab2 3720 . . . . . 6 {𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴
87biantrur 526 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
9 rabun2 3939 . . . . . 6 {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} = ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍})
109sseq1i 3662 . . . . 5 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
116, 8, 103bitr4ri 293 . . . 4 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴)
12 rabss 3712 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴))
13 fvres 6245 . . . . . . . . 9 (𝑎𝐵 → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
1413adantl 481 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
15 simp2r 1108 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝑍𝑉)
16 fvconst2g 6508 . . . . . . . . 9 ((𝑍𝑉𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1715, 16sylan 487 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1814, 17eqeq12d 2666 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹𝑎) = 𝑍))
19 nne 2827 . . . . . . . 8 (¬ (𝐹𝑎) ≠ 𝑍 ↔ (𝐹𝑎) = 𝑍)
2019a1i 11 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (¬ (𝐹𝑎) ≠ 𝑍 ↔ (𝐹𝑎) = 𝑍))
21 id 22 . . . . . . . . 9 (𝑎𝐵𝑎𝐵)
22 simp3 1083 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
23 minel 4066 . . . . . . . . 9 ((𝑎𝐵 ∧ (𝐴𝐵) = ∅) → ¬ 𝑎𝐴)
2421, 22, 23syl2anr 494 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ¬ 𝑎𝐴)
25 mtt 353 . . . . . . . 8 𝑎𝐴 → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2624, 25syl 17 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2718, 20, 263bitr2rd 297 . . . . . 6 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2827ralbidva 3014 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2912, 28syl5bb 272 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
3011, 29syl5bb 272 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
315, 30bitrd 268 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
32 fnfun 6026 . . . . . . 7 (𝐹 Fn (𝐴𝐵) → Fun 𝐹)
33323anim1i 1267 . . . . . 6 ((𝐹 Fn (𝐴𝐵) ∧ 𝐹𝑊𝑍𝑉) → (Fun 𝐹𝐹𝑊𝑍𝑉))
34333expb 1285 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (Fun 𝐹𝐹𝑊𝑍𝑉))
35 suppval1 7346 . . . . 5 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3634, 35syl 17 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
37363adant3 1101 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3837sseq1d 3665 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
39 simp1 1081 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐹 Fn (𝐴𝐵))
40 ssun2 3810 . . . . 5 𝐵 ⊆ (𝐴𝐵)
4140a1i 11 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
42 fnssres 6042 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐹𝐵) Fn 𝐵)
4339, 41, 42syl2anc 694 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹𝐵) Fn 𝐵)
44 fnconstg 6131 . . . . 5 (𝑍𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
4544adantl 481 . . . 4 ((𝐹𝑊𝑍𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
46453ad2ant2 1103 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
47 eqfnfv 6351 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐵 × {𝑍}) Fn 𝐵) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4843, 46, 47syl2anc 694 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4931, 38, 483bitr4d 300 1 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹𝐵) = (𝐵 × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  cres 5145  Fun wfun 5920   Fn wfn 5921  cfv 5926  (class class class)co 6690   supp csupp 7340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341
This theorem is referenced by:  fnsuppeq0  7368  frlmsslss2  20162  resf1o  29633
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