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Theorem fnsuppres 8122
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 6605 . . . . . 6 (𝐹 Fn (𝐴𝐵) → dom 𝐹 = (𝐴𝐵))
21rabeqdv 3422 . . . . 5 (𝐹 Fn (𝐴𝐵) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
323ad2ant1 1133 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} = {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍})
43sseq1d 3975 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ({𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
5 unss 4144 . . . . 5 (({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴) ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍}) ⊆ 𝐴)
6 ssrab2 4037 . . . . . 6 {𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴
76biantrur 531 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ∧ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
8 rabun2 4273 . . . . . 6 {𝑎 ∈ (𝐴𝐵) ∣ (𝐹𝑎) ≠ 𝑍} = ({𝑎𝐴 ∣ (𝐹𝑎) ≠ 𝑍} ∪ {𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍})
98sseq1i 3972 . . . . 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 4029 . . . . 5 ({𝑎𝐵 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴 ↔ ∀𝑎𝐵 ((𝐹𝑎) ≠ 𝑍𝑎𝐴))
12 fvres 6861 . . . . . . . . 9 (𝑎𝐵 → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
1312adantl 482 . . . . . . . 8 (((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐵) → ((𝐹𝐵)‘𝑎) = (𝐹𝑎))
14 simp2r 1200 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝑍𝑉)
15 fvconst2g 7151 . . . . . . . . 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 2947 . . . . . . . 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 1138 . . . . . . . . 9 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
22 minel 4425 . . . . . . . . 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 3172 . . . . 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 6602 . . . . . . 7 (𝐹 Fn (𝐴𝐵) → Fun 𝐹)
32313anim1i 1152 . . . . . 6 ((𝐹 Fn (𝐴𝐵) ∧ 𝐹𝑊𝑍𝑉) → (Fun 𝐹𝐹𝑊𝑍𝑉))
33323expb 1120 . . . . 5 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉)) → (Fun 𝐹𝐹𝑊𝑍𝑉))
34 suppval1 8098 . . . . 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 1132 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹 supp 𝑍) = {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍})
3736sseq1d 3975 . 2 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ {𝑎 ∈ dom 𝐹 ∣ (𝐹𝑎) ≠ 𝑍} ⊆ 𝐴))
38 simp1 1136 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐹 Fn (𝐴𝐵))
39 ssun2 4133 . . . . 5 𝐵 ⊆ (𝐴𝐵)
4039a1i 11 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
41 fnssres 6624 . . . 4 ((𝐹 Fn (𝐴𝐵) ∧ 𝐵 ⊆ (𝐴𝐵)) → (𝐹𝐵) Fn 𝐵)
4238, 40, 41syl2anc 584 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐹𝐵) Fn 𝐵)
43 fnconstg 6730 . . . . 5 (𝑍𝑉 → (𝐵 × {𝑍}) Fn 𝐵)
4443adantl 482 . . . 4 ((𝐹𝑊𝑍𝑉) → (𝐵 × {𝑍}) Fn 𝐵)
45443ad2ant2 1134 . . 3 ((𝐹 Fn (𝐴𝐵) ∧ (𝐹𝑊𝑍𝑉) ∧ (𝐴𝐵) = ∅) → (𝐵 × {𝑍}) Fn 𝐵)
46 eqfnfv 6982 . . 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 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586   × cxp 5631  dom cdm 5633  cres 5635  Fun wfun 6490   Fn wfn 6491  cfv 6496  (class class class)co 7357   supp csupp 8092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-supp 8093
This theorem is referenced by:  fnsuppeq0  8123  frlmsslss2  21181  resf1o  31647
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