Theorem suppiniseg 30529

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 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))

Proof of Theorem suppiniseg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3864	. . . 4 ⊢ (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
2		funfn 6358	. . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3	2	biimpi 219	. . . . . . . . . 10 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
4		elsuppfng 7837	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍)))
5	3, 4	syl3an1 1161	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍)))
6	5	baibd 544	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ≠ 𝑍))
7	6	notbid 322	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ≠ 𝑍))
8		nne 2953	. . . . . . 7 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) = 𝑍)
9	7, 8	syl6bb 291	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) = 𝑍))
10		fvex 6664	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11	10	elsn 4530	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
12	9, 11	bitr4di 293	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ {𝑍}))
13	12	pm5.32da 583	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
14	1, 13	syl5bb 286	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
15	3	3ad2ant1 1131	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
16		elpreima 6812	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
17	15, 16	syl 17	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
18	14, 17	bitr4d 285	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (◡𝐹 “ {𝑍})))
19	18	eqrdv 2757	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2949   ∖ cdif 3851  {csn 4515  ^◡ccnv 5516  dom cdm 5517   “ cima 5520  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328  (class class class)co 7143   supp csupp 7828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-br 5026  df-opab 5088  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-supp 7829

This theorem is referenced by:  fressupp  30531  supppreima  30534