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Theorem suppss3 29368
 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 6620 . 2 (𝐺 supp 𝑍) = ((𝑥𝐴𝐵) supp 𝑍)
3 simpl 473 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4 eldifi 3715 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
54adantl 482 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥𝐴)
6 suppss3.2 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐴)
7 suppss3.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
8 fnex 6441 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
96, 7, 8syl2anc 692 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
10 suppss3.z . . . . . . . . . . . . 13 (𝜑𝑍𝑊)
11 suppimacnv 7258 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
129, 10, 11syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1312eleq2d 2684 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))))
14 elpreima 6298 . . . . . . . . . . . 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 268 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1716baibd 947 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1817notbid 308 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1918biimpd 219 . . . . . . 7 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
2019expimpd 628 . . . . . 6 (𝜑 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
21 eldif 3569 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22 fvex 6163 . . . . . . . 8 (𝐹𝑥) ∈ V
23 eldifsn 4292 . . . . . . . 8 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹𝑥) ∈ V ∧ (𝐹𝑥) ≠ 𝑍))
2422, 23mpbiran 952 . . . . . . 7 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹𝑥) ≠ 𝑍)
2524necon2bbii 2841 . . . . . 6 ((𝐹𝑥) = 𝑍 ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍}))
2620, 21, 253imtr4g 285 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹𝑥) = 𝑍))
2726imp 445 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
28 suppss3.3 . . . 4 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)
293, 5, 27, 28syl3anc 1323 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
3029, 7suppss2 7281 . 2 (𝜑 → ((𝑥𝐴𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
312, 30syl5eqss 3633 1 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  Vcvv 3189   ∖ cdif 3556   ⊆ wss 3559  {csn 4153   ↦ cmpt 4678  ◡ccnv 5078   “ cima 5082   Fn wfn 5847  ‘cfv 5852  (class class class)co 6610   supp csupp 7247 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-supp 7248 This theorem is referenced by:  eulerpartlems  30227
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