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Theorem suppss3 30530
 Description: Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypotheses
Ref Expression
suppss3.1 𝐺 = (𝑥𝐴𝐵)
suppss3.a (𝜑𝐴𝑉)
suppss3.z (𝜑𝑍𝑊)
suppss3.2 (𝜑𝐹 Fn 𝐴)
suppss3.3 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)
Assertion
Ref Expression
suppss3 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑍   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem suppss3
StepHypRef Expression
1 suppss3.1 . . 3 𝐺 = (𝑥𝐴𝐵)
21oveq1i 7155 . 2 (𝐺 supp 𝑍) = ((𝑥𝐴𝐵) supp 𝑍)
3 simpl 486 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4 eldifi 4057 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
54adantl 485 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥𝐴)
6 suppss3.2 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐴)
7 suppss3.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
8 fnex 6967 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
96, 7, 8syl2anc 587 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
10 suppss3.z . . . . . . . . . . . . 13 (𝜑𝑍𝑊)
11 suppimacnv 7839 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
129, 10, 11syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1312eleq2d 2875 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))))
14 elpreima 6815 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹 “ (V ∖ {𝑍})) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
156, 14syl 17 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 “ (V ∖ {𝑍})) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1613, 15bitrd 282 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1716baibd 543 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1817notbid 321 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1918biimpd 232 . . . . . . 7 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
2019expimpd 457 . . . . . 6 (𝜑 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
21 eldif 3893 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22 fvex 6668 . . . . . . . 8 (𝐹𝑥) ∈ V
23 eldifsn 4683 . . . . . . . 8 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹𝑥) ∈ V ∧ (𝐹𝑥) ≠ 𝑍))
2422, 23mpbiran 708 . . . . . . 7 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹𝑥) ≠ 𝑍)
2524necon2bbii 3038 . . . . . 6 ((𝐹𝑥) = 𝑍 ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍}))
2620, 21, 253imtr4g 299 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹𝑥) = 𝑍))
2726imp 410 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
28 suppss3.3 . . . 4 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)
293, 5, 27, 28syl3anc 1368 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
3029, 7suppss2 7865 . 2 (𝜑 → ((𝑥𝐴𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
312, 30eqsstrid 3965 1 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442   ∖ cdif 3880   ⊆ wss 3883  {csn 4528   ↦ cmpt 5114  ◡ccnv 5522   “ cima 5526   Fn wfn 6327  ‘cfv 6332  (class class class)co 7145   supp csupp 7826 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-supp 7827 This theorem is referenced by:  eulerpartlems  31794
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