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Theorem suppiniseg 32695
Description: Relation between the support (𝐹 supp 𝑍) and the initial segment (𝐹 “ {𝑍}). (Contributed by Thierry Arnoux, 25-Jun-2024.)
Assertion
Ref Expression
suppiniseg ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))

Proof of Theorem suppiniseg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 3961 . . . 4 (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
2 funfn 6596 . . . . . . . . . . 11 (Fun 𝐹𝐹 Fn dom 𝐹)
32biimpi 216 . . . . . . . . . 10 (Fun 𝐹𝐹 Fn dom 𝐹)
4 elsuppfng 8194 . . . . . . . . . 10 ((𝐹 Fn dom 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 𝑍)))
53, 4syl3an1 1164 . . . . . . . . 9 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 𝑍)))
65baibd 539 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ≠ 𝑍))
76notbid 318 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ≠ 𝑍))
8 nne 2944 . . . . . . 7 (¬ (𝐹𝑥) ≠ 𝑍 ↔ (𝐹𝑥) = 𝑍)
97, 8bitrdi 287 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) = 𝑍))
10 fvex 6919 . . . . . . 7 (𝐹𝑥) ∈ V
1110elsn 4641 . . . . . 6 ((𝐹𝑥) ∈ {𝑍} ↔ (𝐹𝑥) = 𝑍)
129, 11bitr4di 289 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ {𝑍}))
1312pm5.32da 579 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
141, 13bitrid 283 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
1533ad2ant1 1134 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 Fn dom 𝐹)
16 elpreima 7078 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
1715, 16syl 17 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
1814, 17bitr4d 282 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (𝐹 “ {𝑍})))
1918eqrdv 2735 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {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  cdif 3948  {csn 4626  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555   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-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-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  fressupp  32697  supppreima  32700
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