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Theorem suppfnss 8213
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 1193 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐴𝐵)
2 fndm 6672 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32ad2antrr 726 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 = 𝐴)
4 fndm 6672 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
54ad2antlr 727 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐺 = 𝐵)
61, 3, 53sstr4d 4043 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 ⊆ dom 𝐺)
76adantr 480 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
82eleq2d 2825 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
98ad2antrr 726 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹𝑦𝐴))
10 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐺𝑥) = 𝑍 ↔ (𝐺𝑦) = 𝑍))
11 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑦) = 𝑍))
1210, 11imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) ↔ ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
1312rspcv 3618 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
149, 13biimtrdi 253 . . . . . . . . . 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 417 . . . . . . . 8 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))
1716necon3d 2959 . . . . . . 7 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍))
1817ex 412 . . . . . 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 1110 . . . 4 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ (𝐹𝑦) ≠ 𝑍𝑦 ∈ dom 𝐹) → (𝐺𝑦) ≠ 𝑍)
217, 20rabssrabd 4093 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
22 fnfun 6669 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
2322ad2antrr 726 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐹)
24 simpl 482 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
25 ssexg 5329 . . . . . . . 8 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
26253adant3 1131 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐴 ∈ V)
27 fnex 7237 . . . . . . 7 ((𝐹 Fn 𝐴𝐴 ∈ V) → 𝐹 ∈ V)
2824, 26, 27syl2an 596 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐹 ∈ V)
29 simpr3 1195 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝑍𝑊)
30 suppval1 8190 . . . . . 6 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
3123, 28, 29, 30syl3anc 1370 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
32 fnfun 6669 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
3332ad2antlr 727 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐺)
34 simpr 484 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
35 simp2 1136 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐵𝑉)
36 fnex 7237 . . . . . . 7 ((𝐺 Fn 𝐵𝐵𝑉) → 𝐺 ∈ V)
3734, 35, 36syl2an 596 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐺 ∈ V)
38 suppval1 8190 . . . . . 6 ((Fun 𝐺𝐺 ∈ V ∧ 𝑍𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
3933, 37, 29, 38syl3anc 1370 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4031, 39sseq12d 4029 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4140adantr 480 . . 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 257 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
4342ex 412 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  wss 3963  dom cdm 5689  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  funsssuppss  8214  suppofss1d  8228  suppofss2d  8229  lincresunit2  48324
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