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Theorem fnsuppres 8176
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 6646 . . . . . 6 (𝐹 Fn (𝐴𝐵) → dom 𝐹 = (𝐴𝐵))
21rabeqdv 3441 . . . . 5 (𝐹 Fn (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
323ad2ant1 1130 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
43sseq1d 4008 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
5 unss 4179 . . . . 5 (({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
6 ssrab2 4072 . . . . . 6 {𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴
76biantrur 530 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
8 rabun2 4308 . . . . . 6 {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} = ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍})
98sseq1i 4005 . . . . 5 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
105, 7, 93bitr4ri 304 . . . 4 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴)
11 rabss 4064 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴))
12 fvres 6904 . . . . . . . . 9 (𝑎𝐵 → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
1312adantl 481 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
14 simp2r 1197 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝑍𝑉)
15 fvconst2g 7199 . . . . . . . . 9 ((𝑍𝑉𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1614, 15sylan 579 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1713, 16eqeq12d 2742 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹𝑎) = 𝑍))
18 nne 2938 . . . . . . . 8 (¬ (𝐹𝑎) ≠ 𝑍 ↔ (𝐹𝑎) = 𝑍)
1918a1i 11 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (¬ (𝐹𝑎) ≠ 𝑍 ↔ (𝐹𝑎) = 𝑍))
20 id 22 . . . . . . . . 9 (𝑎𝐵𝑎𝐵)
21 simp3 1135 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
22 minel 4460 . . . . . . . . 9 ((𝑎𝐵 ∧ (𝐴𝐵) = ∅) → ¬ 𝑎𝐴)
2320, 21, 22syl2anr 596 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ¬ 𝑎𝐴)
24 mtt 364 . . . . . . . 8 𝑎𝐴 → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2523, 24syl 17 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2617, 19, 253bitr2rd 308 . . . . . 6 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2726ralbidva 3169 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2811, 27bitrid 283 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2910, 28bitrid 283 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
304, 29bitrd 279 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
31 fnfun 6643 . . . . . . 7 (𝐹 Fn (𝐴𝐵) → Fun 𝐹)
32313anim1i 1149 . . . . . 6 ((𝐹 Fn (𝐴𝐵) ∧ 𝐹𝑊𝑍𝑉) → (Fun 𝐹𝐹𝑊𝑍𝑉))
33323expb 1117 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (Fun 𝐹𝐹𝑊𝑍𝑉))
34 suppval1 8152 . . . . 5 ((Fun 𝐹𝐹𝑊𝑍𝑉) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3533, 34syl 17 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
36353adant3 1129 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3736sseq1d 4008 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
38 simp1 1133 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐹 Fn (𝐴𝐵))
39 ssun2 4168 . . . . 5 𝐵 ⊆ (𝐴𝐵)
4039a1i 11 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
41 fnssres 6667 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐹𝐵) Fn 𝐵)
4238, 40, 41syl2anc 583 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹𝐵) Fn 𝐵)
43 fnconstg 6773 . . . . 5 (𝑍𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
4443adantl 481 . . . 4 ((𝐹𝑊𝑍𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45443ad2ant2 1131 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46 eqfnfv 7026 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐵 × {𝑍}) Fn 𝐵) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4742, 45, 46syl2anc 583 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4830, 37, 473bitr4d 311 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wral 3055  {crab 3426  cun 3941  cin 3942  wss 3943  c0 4317  {csn 4623   × cxp 5667  dom cdm 5669  cres 5671  Fun wfun 6531   Fn wfn 6532  cfv 6537  (class class class)co 7405   supp csupp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8147
This theorem is referenced by:  fnsuppeq0  8177  frlmsslss2  21670  resf1o  32462
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