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Theorem suppfnss 8170
Description: The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)
Assertion
Ref Expression
suppfnss (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝑍
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem suppfnss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr1 1194 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐴𝐵)
2 fndm 6649 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32ad2antrr 724 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 = 𝐴)
4 fndm 6649 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
54ad2antlr 725 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐺 = 𝐵)
61, 3, 53sstr4d 4028 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 ⊆ dom 𝐺)
76adantr 481 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
82eleq2d 2819 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
98ad2antrr 724 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹𝑦𝐴))
10 fveqeq2 6897 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐺𝑥) = 𝑍 ↔ (𝐺𝑦) = 𝑍))
11 fveqeq2 6897 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑦) = 𝑍))
1210, 11imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) ↔ ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
1312rspcv 3608 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
149, 13syl6bi 252 . . . . . . . . . 10 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1514com23 86 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝑦 ∈ dom 𝐹 → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1615imp31 418 . . . . . . . 8 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))
1716necon3d 2961 . . . . . . 7 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍))
1817ex 413 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝑦 ∈ dom 𝐹 → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍)))
1918com23 86 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → ((𝐹𝑦) ≠ 𝑍 → (𝑦 ∈ dom 𝐹 → (𝐺𝑦) ≠ 𝑍)))
20193imp 1111 . . . 4 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ (𝐹𝑦) ≠ 𝑍𝑦 ∈ dom 𝐹) → (𝐺𝑦) ≠ 𝑍)
217, 20rabssrabd 4080 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
22 fnfun 6646 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
2322ad2antrr 724 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐹)
24 simpl 483 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
25 ssexg 5322 . . . . . . . 8 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
26253adant3 1132 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐴 ∈ V)
27 fnex 7215 . . . . . . 7 ((𝐹 Fn 𝐴𝐴 ∈ V) → 𝐹 ∈ V)
2824, 26, 27syl2an 596 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐹 ∈ V)
29 simpr3 1196 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝑍𝑊)
30 suppval1 8148 . . . . . 6 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
3123, 28, 29, 30syl3anc 1371 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
32 fnfun 6646 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
3332ad2antlr 725 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐺)
34 simpr 485 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
35 simp2 1137 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐵𝑉)
36 fnex 7215 . . . . . . 7 ((𝐺 Fn 𝐵𝐵𝑉) → 𝐺 ∈ V)
3734, 35, 36syl2an 596 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐺 ∈ V)
38 suppval1 8148 . . . . . 6 ((Fun 𝐺𝐺 ∈ V ∧ 𝑍𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
3933, 37, 29, 38syl3anc 1371 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4031, 39sseq12d 4014 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4140adantr 481 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4221, 41mpbird 256 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
4342ex 413 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  {crab 3432  Vcvv 3474  wss 3947  dom cdm 5675  Fun wfun 6534   Fn wfn 6535  cfv 6540  (class class class)co 7405   supp csupp 8142
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8143
This theorem is referenced by:  funsssuppss  8171  suppofss1d  8185  suppofss2d  8186  lincresunit2  47112
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