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Theorem suppss3 32638
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 7426 . 2 (𝐺 supp 𝑍) = ((𝑥𝐴𝐵) supp 𝑍)
3 simpl 481 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4 eldifi 4123 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
54adantl 480 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥𝐴)
6 suppss3.2 . . . . . . . . . . . . . 14 (𝜑𝐹 Fn 𝐴)
7 suppss3.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
8 fnex 7226 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
96, 7, 8syl2anc 582 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
10 suppss3.z . . . . . . . . . . . . 13 (𝜑𝑍𝑊)
11 suppimacnv 8180 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
129, 10, 11syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1312eleq2d 2812 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (𝐹 “ (V ∖ {𝑍}))))
14 elpreima 7063 . . . . . . . . . . . 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 278 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ (V ∖ {𝑍}))))
1716baibd 538 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1817notbid 317 . . . . . . . 8 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
1918biimpd 228 . . . . . . 7 ((𝜑𝑥𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
2019expimpd 452 . . . . . 6 (𝜑 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹𝑥) ∈ (V ∖ {𝑍})))
21 eldif 3956 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22 fvex 6906 . . . . . . . 8 (𝐹𝑥) ∈ V
23 eldifsn 4785 . . . . . . . 8 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹𝑥) ∈ V ∧ (𝐹𝑥) ≠ 𝑍))
2422, 23mpbiran 707 . . . . . . 7 ((𝐹𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹𝑥) ≠ 𝑍)
2524necon2bbii 2982 . . . . . 6 ((𝐹𝑥) = 𝑍 ↔ ¬ (𝐹𝑥) ∈ (V ∖ {𝑍}))
2620, 21, 253imtr4g 295 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹𝑥) = 𝑍))
2726imp 405 . . . 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 8207 . 2 (𝜑 → ((𝑥𝐴𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
312, 30eqsstrid 4027 1 (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  Vcvv 3462  cdif 3943  wss 3946  {csn 4623  cmpt 5228  ccnv 5673  cima 5677   Fn wfn 6541  cfv 6546  (class class class)co 7416   supp csupp 8166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-supp 8167
This theorem is referenced by:  evls1fldgencl  33562  eulerpartlems  34207
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