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Theorem suppiniseg 30449
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 3894 . . . 4 (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
2 funfn 6358 . . . . . . . . . . . 12 (Fun 𝐹𝐹 Fn dom 𝐹)
32biimpi 219 . . . . . . . . . . 11 (Fun 𝐹𝐹 Fn dom 𝐹)
433ad2ant1 1130 . . . . . . . . . 10 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝐹 Fn dom 𝐹)
5 dmexg 7598 . . . . . . . . . . 11 (𝐹𝑉 → dom 𝐹 ∈ V)
653ad2ant2 1131 . . . . . . . . . 10 ((Fun 𝐹𝐹𝑉𝑍𝑊) → dom 𝐹 ∈ V)
7 simp3 1135 . . . . . . . . . 10 ((Fun 𝐹𝐹𝑉𝑍𝑊) → 𝑍𝑊)
8 elsuppfn 7825 . . . . . . . . . 10 ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ V ∧ 𝑍𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 𝑍)))
94, 6, 7, 8syl3anc 1368 . . . . . . . . 9 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ≠ 𝑍)))
109baibd 543 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ≠ 𝑍))
1110notbid 321 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹𝑥) ≠ 𝑍))
12 nne 2994 . . . . . . 7 (¬ (𝐹𝑥) ≠ 𝑍 ↔ (𝐹𝑥) = 𝑍)
1311, 12syl6bb 290 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) = 𝑍))
14 fvex 6662 . . . . . . 7 (𝐹𝑥) ∈ V
1514elsn 4543 . . . . . 6 ((𝐹𝑥) ∈ {𝑍} ↔ (𝐹𝑥) = 𝑍)
1613, 15syl6bbr 292 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹𝑥) ∈ {𝑍}))
1716pm5.32da 582 . . . 4 ((Fun 𝐹𝐹𝑉𝑍𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
181, 17syl5bb 286 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
19 elpreima 6809 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
204, 19syl 17 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ {𝑍})))
2118, 20bitr4d 285 . 2 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (𝐹 “ {𝑍})))
2221eqrdv 2799 1 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐹 “ {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  Vcvv 3444  cdif 3881  {csn 4528  ccnv 5522  dom cdm 5523  cima 5526  Fun wfun 6322   Fn wfn 6323  cfv 6328  (class class class)co 7139   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818
This theorem is referenced by:  fressupp  30451  supppreima  30454
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