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Theorem fnsuppres 7715
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 6332 . . . . . 6 (𝐹 Fn (𝐴𝐵) → dom 𝐹 = (𝐴𝐵))
21rabeqdv 3432 . . . . 5 (𝐹 Fn (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
323ad2ant1 1126 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
43sseq1d 3925 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
5 unss 4087 . . . . 5 (({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
6 ssrab2 3983 . . . . . 6 {𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴
76biantrur 531 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
8 rabun2 4208 . . . . . 6 {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} = ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍})
98sseq1i 3922 . . . . 5 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
105, 7, 93bitr4ri 305 . . . 4 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴)
11 rabss 3975 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴))
12 fvres 6564 . . . . . . . . 9 (𝑎𝐵 → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
1312adantl 482 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
14 simp2r 1193 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝑍𝑉)
15 fvconst2g 6838 . . . . . . . . 9 ((𝑍𝑉𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1614, 15sylan 580 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1713, 16eqeq12d 2812 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹𝑎) = 𝑍))
18 nne 2990 . . . . . . . 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 1131 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
22 minel 4335 . . . . . . . . 9 ((𝑎𝐵 ∧ (𝐴𝐵) = ∅) → ¬ 𝑎𝐴)
2320, 21, 22syl2anr 596 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ¬ 𝑎𝐴)
24 mtt 366 . . . . . . . 8 𝑎𝐴 → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2523, 24syl 17 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (¬ (𝐹𝑎) ≠ 𝑍 ↔ ((𝐹𝑎) ≠ 𝑍𝑎𝐴)))
2617, 19, 253bitr2rd 309 . . . . . 6 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2726ralbidva 3165 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2811, 27syl5bb 284 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2910, 28syl5bb 284 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
304, 29bitrd 280 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
31 fnfun 6330 . . . . . . 7 (𝐹 Fn (𝐴𝐵) → Fun 𝐹)
32313anim1i 1145 . . . . . 6 ((𝐹 Fn (𝐴𝐵) ∧ 𝐹𝑊𝑍𝑉) → (Fun 𝐹𝐹𝑊𝑍𝑉))
33323expb 1113 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (Fun 𝐹𝐹𝑊𝑍𝑉))
34 suppval1 7694 . . . . 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 1125 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3736sseq1d 3925 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
38 simp1 1129 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐹 Fn (𝐴𝐵))
39 ssun2 4076 . . . . 5 𝐵 ⊆ (𝐴𝐵)
4039a1i 11 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
41 fnssres 6347 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐹𝐵) Fn 𝐵)
4238, 40, 41syl2anc 584 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹𝐵) Fn 𝐵)
43 fnconstg 6442 . . . . 5 (𝑍𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
4443adantl 482 . . . 4 ((𝐹𝑊𝑍𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45443ad2ant2 1127 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46 eqfnfv 6674 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐵 × {𝑍}) Fn 𝐵) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4742, 45, 46syl2anc 584 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹𝐵) = (𝐵 × {𝑍}) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
4830, 37, 473bitr4d 312 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  wral 3107  {crab 3111  cun 3863  cin 3864  wss 3865  c0 4217  {csn 4478   × cxp 5448  dom cdm 5450  cres 5452  Fun wfun 6226   Fn wfn 6227  cfv 6232  (class class class)co 7023   supp csupp 7688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-supp 7689
This theorem is referenced by:  fnsuppeq0  7716  frlmsslss2  20605  resf1o  30150
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