Theorem suppiniseg 32698

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 3986	. . . 4 ⊢ (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
2		funfn 6608	. . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3	2	biimpi 216	. . . . . . . . . 10 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
4		elsuppfng 8210	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍)))
5	3, 4	syl3an1 1163	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍)))
6	5	baibd 539	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ≠ 𝑍))
7	6	notbid 318	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ≠ 𝑍))
8		nne 2950	. . . . . . 7 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) = 𝑍)
9	7, 8	bitrdi 287	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) = 𝑍))
10		fvex 6933	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11	10	elsn 4663	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
12	9, 11	bitr4di 289	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ {𝑍}))
13	12	pm5.32da 578	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
14	1, 13	bitrid 283	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍})))
15	3	3ad2ant1 1133	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹)
16		elpreima 7091	. . . 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 282	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (◡𝐹 “ {𝑍})))
19	18	eqrdv 2738	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973  {csn 4648  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by:  fressupp  32700  supppreima  32703