Theorem suppcofn 6500

Description: The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) (Revised by SN, 15-Sep-2023.)

Assertion

Ref	Expression
suppcofn	⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

Proof of Theorem suppcofn

Dummy variables 𝑥 𝑓 𝑖 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6470	. . . . 5 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑓 ∣ (𝑓 “ {𝑖}) ≠ {𝑧}})
2	1	elmpocl2 6280	. . . 4 ⊢ (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) → 𝑍 ∈ V)
3	2	a1i 9	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) → 𝑍 ∈ V))
4		simprr 537	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → Fun 𝐺)
5	4	funfnd 5406	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → 𝐺 Fn dom 𝐺)
6		elpreima 5822	. . . . . . 7 ⊢ (𝐺 Fn dom 𝐺 → (𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐺 ∧ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍))))
7	5, 6	syl 14	. . . . . 6 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐺 ∧ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍))))
8	7	simplbda 384	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍))) → (𝐺‘𝑥) ∈ (𝐹 supp 𝑍))
9	1	elmpocl2 6280	. . . . 5 ⊢ ((𝐺‘𝑥) ∈ (𝐹 supp 𝑍) → 𝑍 ∈ V)
10	8, 9	syl 14	. . . 4 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍))) → 𝑍 ∈ V)
11	10	ex 115	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍)) → 𝑍 ∈ V))
12		funco 5415	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
13	12	adantl 277	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → Fun (𝐹 ∘ 𝐺))
14	13	funfnd 5406	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺))
15	14	adantr 276	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺))
16		coexg 5330	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
17	16	ad2antrr 492	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (𝐹 ∘ 𝐺) ∈ V)
18		simpr 110	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
19		suppimacnvfn 6480	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐺) Fn dom (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
20	15, 17, 18, 19	syl3anc 1278	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
21		cnvco 4963	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
22	21	imaeq1i 5121	. . . . . . 7 ⊢ (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍}))
23	22	a1i 9	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})))
24		imaco 5291	. . . . . . 7 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
25		simprl 535	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → Fun 𝐹)
26	25	funfnd 5406	. . . . . . . . . 10 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → 𝐹 Fn dom 𝐹)
27	26	adantr 276	. . . . . . . . 9 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → 𝐹 Fn dom 𝐹)
28		simplll 539	. . . . . . . . 9 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → 𝐹 ∈ 𝑉)
29		suppimacnvfn 6480	. . . . . . . . 9 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
30	27, 28, 18, 29	syl3anc 1278	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
31	30	imaeq2d 5124	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
32	24, 31	eqtr4id 2290	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
33	20, 23, 32	3eqtrd 2275	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
34	33	eleq2d 2308	. . . 4 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) ↔ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍))))
35	34	ex 115	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑍 ∈ V → (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) ↔ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍)))))
36	3, 11, 35	pm5.21ndd 717	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) ↔ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍))))
37	36	eqrdv 2236	1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  {crab 2532  Vcvv 2821   ∖ cdif 3217  {csn 3708  ^◡ccnv 4771  dom cdm 4772   “ cima 4775   ∘ ccom 4776  Fun wfun 5369   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:  supp0cosupp0fn  6501  imacosuppfn  6502  fsuppcorn  7295