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Theorem suppss3 30212
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 6986 . 2 (𝐺 supp 𝑍) = ((𝑥𝐴𝐵) supp 𝑍)
3 simpl 475 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4 eldifi 3994 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
54adantl 474 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥𝐴)
6 suppss3.2 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐴)
7 suppss3.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
8 fnex 6806 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
96, 7, 8syl2anc 576 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
10 suppss3.z . . . . . . . . . . . . 13 (𝜑𝑍𝑊)
11 suppimacnv 7644 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
129, 10, 11syl2anc 576 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1312eleq2d 2852 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))))
14 elpreima 6653 . . . . . . . . . . . 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 271 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1716baibd 532 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1817notbid 310 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1918biimpd 221 . . . . . . 7 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
2019expimpd 446 . . . . . 6 (𝜑 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
21 eldif 3840 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22 fvex 6512 . . . . . . . 8 (𝐹𝑥) ∈ V
23 eldifsn 4593 . . . . . . . 8 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹𝑥) ∈ V ∧ (𝐹𝑥) ≠ 𝑍))
2422, 23mpbiran 696 . . . . . . 7 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹𝑥) ≠ 𝑍)
2524necon2bbii 3019 . . . . . 6 ((𝐹𝑥) = 𝑍 ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍}))
2620, 21, 253imtr4g 288 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹𝑥) = 𝑍))
2726imp 398 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
28 suppss3.3 . . . 4 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)
293, 5, 27, 28syl3anc 1351 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
3029, 7suppss2 7667 . 2 (𝜑 → ((𝑥𝐴𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
312, 30syl5eqss 3906 1 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2968  Vcvv 3416  cdif 3827  wss 3830  {csn 4441  cmpt 5008  ccnv 5406  cima 5410   Fn wfn 6183  cfv 6188  (class class class)co 6976   supp csupp 7633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-supp 7634
This theorem is referenced by:  eulerpartlems  31260
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