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Theorem suppfnss 7490
 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 1234 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐴𝐵)
2 fndm 6152 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32ad2antrr 764 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 = 𝐴)
4 fndm 6152 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
54ad2antlr 765 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐺 = 𝐵)
61, 3, 53sstr4d 3790 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → dom 𝐹 ⊆ dom 𝐺)
76adantr 472 . . . 4 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
82eleq2d 2826 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹𝑦𝐴))
98ad2antrr 764 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹𝑦𝐴))
10 fveq2 6354 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
1110eqeq1d 2763 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐺𝑥) = 𝑍 ↔ (𝐺𝑦) = 𝑍))
12 fveq2 6354 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1312eqeq1d 2763 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑦) = 𝑍))
1411, 13imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) ↔ ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
1514rspcv 3446 . . . . . . . . . . 11 (𝑦𝐴 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍)))
169, 15syl6bi 243 . . . . . . . . . 10 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝑦 ∈ dom 𝐹 → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1716com23 86 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝑦 ∈ dom 𝐹 → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))))
1817imp31 447 . . . . . . . 8 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺𝑦) = 𝑍 → (𝐹𝑦) = 𝑍))
1918necon3d 2954 . . . . . . 7 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍))
2019ex 449 . . . . . 6 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝑦 ∈ dom 𝐹 → ((𝐹𝑦) ≠ 𝑍 → (𝐺𝑦) ≠ 𝑍)))
2120com23 86 . . . . 5 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → ((𝐹𝑦) ≠ 𝑍 → (𝑦 ∈ dom 𝐹 → (𝐺𝑦) ≠ 𝑍)))
22213imp 1102 . . . 4 (((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) ∧ (𝐹𝑦) ≠ 𝑍𝑦 ∈ dom 𝐹) → (𝐺𝑦) ≠ 𝑍)
237, 22rabssrabd 3831 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
24 fnfun 6150 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
2524ad2antrr 764 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐹)
26 simpl 474 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
27 ssexg 4957 . . . . . . . 8 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
28273adant3 1127 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐴 ∈ V)
29 fnex 6647 . . . . . . 7 ((𝐹 Fn 𝐴𝐴 ∈ V) → 𝐹 ∈ V)
3026, 28, 29syl2an 495 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐹 ∈ V)
31 simpr3 1238 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝑍𝑊)
32 suppval1 7471 . . . . . 6 ((Fun 𝐹𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
3325, 30, 31, 32syl3anc 1477 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍})
34 fnfun 6150 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
3534ad2antlr 765 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → Fun 𝐺)
36 simpr 479 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
37 simp2 1132 . . . . . . 7 ((𝐴𝐵𝐵𝑉𝑍𝑊) → 𝐵𝑉)
38 fnex 6647 . . . . . . 7 ((𝐺 Fn 𝐵𝐵𝑉) → 𝐺 ∈ V)
3936, 37, 38syl2an 495 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → 𝐺 ∈ V)
40 suppval1 7471 . . . . . 6 ((Fun 𝐺𝐺 ∈ V ∧ 𝑍𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4135, 39, 31, 40syl3anc 1477 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍})
4233, 41sseq12d 3776 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺𝑦) ≠ 𝑍}))
4342adantr 472 . . 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}))
4423, 43mpbird 247 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵𝐵𝑉𝑍𝑊)) ∧ ∀𝑥𝐴 ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
4544ex 449 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140   ≠ wne 2933  ∀wral 3051  {crab 3055  Vcvv 3341   ⊆ wss 3716  dom cdm 5267  Fun wfun 6044   Fn wfn 6045  ‘cfv 6050  (class class class)co 6815   supp csupp 7465 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-supp 7466 This theorem is referenced by:  funsssuppss  7492  suppofss1d  7503  suppofss2d  7504  lincresunit2  42796
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