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Theorem fnsuppres 8078
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 6589 . . . . . 6 (𝐹 Fn (𝐴𝐵) → dom 𝐹 = (𝐴𝐵))
21rabeqdv 3418 . . . . 5 (𝐹 Fn (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
323ad2ant1 1132 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
43sseq1d 3963 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
5 unss 4132 . . . . 5 (({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
6 ssrab2 4025 . . . . . 6 {𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴
76biantrur 531 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
8 rabun2 4261 . . . . . 6 {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} = ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍})
98sseq1i 3960 . . . . 5 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
105, 7, 93bitr4ri 303 . . . 4 ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴)
11 rabss 4017 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴))
12 fvres 6845 . . . . . . . . 9 (𝑎𝐵 → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
1312adantl 482 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
14 simp2r 1199 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝑍𝑉)
15 fvconst2g 7134 . . . . . . . . 9 ((𝑍𝑉𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1614, 15sylan 580 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐵 × {𝑍})‘𝑎) = 𝑍)
1713, 16eqeq12d 2752 . . . . . . 7 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎) ↔ (𝐹𝑎) = 𝑍))
18 nne 2944 . . . . . . . 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 1137 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
22 minel 4413 . . . . . . . . 9 ((𝑎𝐵 ∧ (𝐴𝐵) = ∅) → ¬ 𝑎𝐴)
2320, 21, 22syl2anr 597 . . . . . . . 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 307 . . . . . 6 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → (((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2726ralbidva 3168 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴) ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2811, 27bitrid 282 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
2910, 28bitrid 282 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
304, 29bitrd 278 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝐵)‘𝑎) = ((𝐵 × {𝑍})‘𝑎)))
31 fnfun 6586 . . . . . . 7 (𝐹 Fn (𝐴𝐵) → Fun 𝐹)
32313anim1i 1151 . . . . . 6 ((𝐹 Fn (𝐴𝐵) ∧ 𝐹𝑊𝑍𝑉) → (Fun 𝐹𝐹𝑊𝑍𝑉))
33323expb 1119 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (Fun 𝐹𝐹𝑊𝑍𝑉))
34 suppval1 8054 . . . . 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 1131 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3736sseq1d 3963 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
38 simp1 1135 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐹 Fn (𝐴𝐵))
39 ssun2 4121 . . . . 5 𝐵 ⊆ (𝐴𝐵)
4039a1i 11 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
41 fnssres 6608 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐹𝐵) Fn 𝐵)
4238, 40, 41syl2anc 584 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹𝐵) Fn 𝐵)
43 fnconstg 6714 . . . . 5 (𝑍𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
4443adantl 482 . . . 4 ((𝐹𝑊𝑍𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45443ad2ant2 1133 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46 eqfnfv 6966 . . 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 310 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 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940  wral 3061  {crab 3403  cun 3896  cin 3897  wss 3898  c0 4270  {csn 4574   × cxp 5619  dom cdm 5621  cres 5623  Fun wfun 6474   Fn wfn 6475  cfv 6480  (class class class)co 7338   supp csupp 8048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-fv 6488  df-ov 7341  df-oprab 7342  df-mpo 7343  df-supp 8049
This theorem is referenced by:  fnsuppeq0  8079  frlmsslss2  21089  resf1o  31352
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