Theorem suppimacnvfn 6480

Description: Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)

Assertion

Ref	Expression
suppimacnvfn	⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))

Proof of Theorem suppimacnvfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 ∈ 𝑉)
2		vex 2824	. . . . . . . . 9 ⊢ 𝑥 ∈ V
3		fvexg 5712	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝐹‘𝑥) ∈ V)
4	1, 2, 3	sylancl 417	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹‘𝑥) ∈ V)
5		elsng 3723	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍))
6	4, 5	syl 14	. . . . . . 7 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍))
7	6	necon3bbid 2460	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (¬ (𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) ≠ 𝑍))
8	4	biantrurd 305	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (¬ (𝐹‘𝑥) ∈ {𝑍} ↔ ((𝐹‘𝑥) ∈ V ∧ ¬ (𝐹‘𝑥) ∈ {𝑍})))
9	7, 8	bitr3d 190	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹‘𝑥) ≠ 𝑍 ↔ ((𝐹‘𝑥) ∈ V ∧ ¬ (𝐹‘𝑥) ∈ {𝑍})))
10		eldif 3229	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ ¬ (𝐹‘𝑥) ∈ {𝑍}))
11	9, 10	bitr4di 198	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
12	11	anbi2d 468	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ≠ 𝑍) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
13		elsuppfng 6476	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ≠ 𝑍)))
14		elpreima 5822	. . . 4 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
15	14	3ad2ant1 1049	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
16	12, 13, 15	3bitr4d 220	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
17	16	eqrdv 2236	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  Vcvv 2821   ∖ cdif 3217  {csn 3708  ^◡ccnv 4771   “ cima 4775   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   supp csupp 6469

This theorem was proved from 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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  fsuppeq  6481  fsuppeqg  6482  mptsuppdifd  6489  suppcofn  6500