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Theorem suppfnss 8121
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 1195 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐴𝐵)
2 fndm 6606 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32ad2antrr 725 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 = 𝐴)
4 fndm 6606 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
54ad2antlr 726 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐺 = 𝐵)
61, 3, 53sstr4d 3992 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 ⊆ dom 𝐺)
76adantr 482 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
82eleq2d 2824 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
98ad2antrr 725 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹𝑦𝐴))
10 fveqeq2 6852 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐺𝑥) = 𝑍 ↔ (𝐺𝑦) = 𝑍))
11 fveqeq2 6852 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑦) = 𝑍))
1210, 11imbi12d 345 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) ↔ ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
1312rspcv 3578 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
149, 13syl6bi 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 419 . . . . . . . 8 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))
1716necon3d 2965 . . . . . . 7 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍))
1817ex 414 . . . . . 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 1112 . . . 4 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ (𝐹𝑦) ≠ 𝑍𝑦 ∈ dom 𝐹) → (𝐺𝑦) ≠ 𝑍)
217, 20rabssrabd 4042 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
22 fnfun 6603 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
2322ad2antrr 725 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐹)
24 simpl 484 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
25 ssexg 5281 . . . . . . . 8 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
26253adant3 1133 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐴 ∈ V)
27 fnex 7168 . . . . . . 7 ((𝐹 Fn 𝐴𝐴 ∈ V) → 𝐹 ∈ V)
2824, 26, 27syl2an 597 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐹 ∈ V)
29 simpr3 1197 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝑍𝑊)
30 suppval1 8099 . . . . . 6 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
3123, 28, 29, 30syl3anc 1372 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
32 fnfun 6603 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
3332ad2antlr 726 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐺)
34 simpr 486 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
35 simp2 1138 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐵𝑉)
36 fnex 7168 . . . . . . 7 ((𝐺 Fn 𝐵𝐵𝑉) → 𝐺 ∈ V)
3734, 35, 36syl2an 597 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐺 ∈ V)
38 suppval1 8099 . . . . . 6 ((Fun 𝐺𝐺 ∈ V ∧ 𝑍𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
3933, 37, 29, 38syl3anc 1372 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4031, 39sseq12d 3978 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4140adantr 482 . . 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 414 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  {crab 3408  Vcvv 3446  wss 3911  dom cdm 5634  Fun wfun 6491   Fn wfn 6492  cfv 6497  (class class class)co 7358   supp csupp 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8094
This theorem is referenced by:  funsssuppss  8122  suppofss1d  8136  suppofss2d  8137  lincresunit2  46566
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