Theorem suppinisegdc 6513

Description: Relation between the support (𝐹 supp 𝑍) and the initial segment (^◡𝐹 “ {𝑍}). (Contributed by Thierry Arnoux, 25-Jun-2024.) (Revised by Jim Kingdon, 18-Oct-2026.)

Hypotheses

Ref	Expression
suppinisegdc.fun	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppinisegdc.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
suppinisegdc.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
suppinisegdc.dc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)

Assertion

Ref	Expression
suppinisegdc	⊢ (𝜑 → (𝐴 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑦,𝑍   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑍(𝑥)

Proof of Theorem suppinisegdc

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3229	. . . 4 ⊢ (𝑤 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑤 ∈ 𝐴 ∧ ¬ 𝑤 ∈ (𝐹 supp 𝑍)))
2		eqeq2 2248	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝐹‘𝑤) = 𝑦 ↔ (𝐹‘𝑤) = 𝑍))
3	2	dcbid 850	. . . . . . . 8 ⊢ (𝑦 = 𝑍 → (DECID (𝐹‘𝑤) = 𝑦 ↔ DECID (𝐹‘𝑤) = 𝑍))
4		eqeq1 2245	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥 = 𝑦 ↔ (𝐹‘𝑤) = 𝑦))
5	4	dcbid 850	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑤) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘𝑤) = 𝑦))
6	5	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑤) → (∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐵 DECID (𝐹‘𝑤) = 𝑦))
7		suppinisegdc.dc	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
8	7	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
9		suppinisegdc.fun	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
10	9	ffvelcdmda 5846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐵)
11	6, 8, 10	rspcdva 2934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 DECID (𝐹‘𝑤) = 𝑦)
12		suppinisegdc.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13	12	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑍 ∈ 𝐵)
14	3, 11, 13	rspcdva 2934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → DECID (𝐹‘𝑤) = 𝑍)
15		nnedc 2425	. . . . . . 7 ⊢ (DECID (𝐹‘𝑤) = 𝑍 → (¬ (𝐹‘𝑤) ≠ 𝑍 ↔ (𝐹‘𝑤) = 𝑍))
16	14, 15	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ (𝐹‘𝑤) ≠ 𝑍 ↔ (𝐹‘𝑤) = 𝑍))
17	9	ffnd 5537	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18		suppinisegdc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
19		elsuppfng 6486	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑤 ∈ (𝐹 supp 𝑍) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ≠ 𝑍)))
20	17, 18, 12, 19	syl3anc 1278	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ (𝐹 supp 𝑍) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ≠ 𝑍)))
21	20	baibd 935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑤) ≠ 𝑍))
22	21	notbid 677	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑤) ≠ 𝑍))
23	18	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝐹 ∈ 𝑉)
24		vex 2824	. . . . . . . 8 ⊢ 𝑤 ∈ V
25		fvexg 5717	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑤 ∈ V) → (𝐹‘𝑤) ∈ V)
26	23, 24, 25	sylancl 417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ V)
27		elsng 3724	. . . . . . 7 ⊢ ((𝐹‘𝑤) ∈ V → ((𝐹‘𝑤) ∈ {𝑍} ↔ (𝐹‘𝑤) = 𝑍))
28	26, 27	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤) ∈ {𝑍} ↔ (𝐹‘𝑤) = 𝑍))
29	16, 22, 28	3bitr4d 220	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑤) ∈ {𝑍}))
30	29	pm5.32da 456	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ 𝐴 ∧ ¬ 𝑤 ∈ (𝐹 supp 𝑍)) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ {𝑍})))
31	1, 30	bitrid 192	. . 3 ⊢ (𝜑 → (𝑤 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ {𝑍})))
32		elpreima 5831	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ {𝑍})))
33	17, 32	syl 14	. . 3 ⊢ (𝜑 → (𝑤 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ {𝑍})))
34	31, 33	bitr4d 191	. 2 ⊢ (𝜑 → (𝑤 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ 𝑤 ∈ (◡𝐹 “ {𝑍})))
35	34	eqrdv 2236	1 ⊢ (𝜑 → (𝐴 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  Vcvv 2821   ∖ cdif 3217  {csn 3709  ^◡ccnv 4775   “ cima 4779   Fn wfn 5375  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   supp csupp 6479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480

This theorem is used by:  funisfsuppdc  7298