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Theorem suppss3 32735
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 7441 . 2 (𝐺 supp 𝑍) = ((𝑥𝐴𝐵) supp 𝑍)
3 simpl 482 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4 eldifi 4131 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
54adantl 481 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥𝐴)
6 suppss3.2 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐴)
7 suppss3.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
8 fnex 7237 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
96, 7, 8syl2anc 584 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
10 suppss3.z . . . . . . . . . . . . 13 (𝜑𝑍𝑊)
11 suppimacnv 8199 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
129, 10, 11syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1312eleq2d 2827 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))))
14 elpreima 7078 . . . . . . . . . . . 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 279 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1716baibd 539 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1817notbid 318 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1918biimpd 229 . . . . . . 7 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
2019expimpd 453 . . . . . 6 (𝜑 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
21 eldif 3961 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22 fvex 6919 . . . . . . . 8 (𝐹𝑥) ∈ V
23 eldifsn 4786 . . . . . . . 8 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹𝑥) ∈ V ∧ (𝐹𝑥) ≠ 𝑍))
2422, 23mpbiran 709 . . . . . . 7 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹𝑥) ≠ 𝑍)
2524necon2bbii 2992 . . . . . 6 ((𝐹𝑥) = 𝑍 ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍}))
2620, 21, 253imtr4g 296 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹𝑥) = 𝑍))
2726imp 406 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
28 suppss3.3 . . . 4 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)
293, 5, 27, 28syl3anc 1373 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
3029, 7suppss2 8225 . 2 (𝜑 → ((𝑥𝐴𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
312, 30eqsstrid 4022 1 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  wss 3951  {csn 4626  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  cfv 6561  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  evls1fldgencl  33720  eulerpartlems  34362
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