Theorem suppcofn 6444

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 6414	. . . . 5 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑓 ∣ (𝑓 “ {𝑖}) ≠ {𝑧}})
2	1	elmpocl2 6229	. . . 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 533	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → Fun 𝐺)
5	4	funfnd 5364	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → 𝐺 Fn dom 𝐺)
6		elpreima 5775	. . . . . . 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 6229	. . . . 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 5373	. . . . . . . . . 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 5364	. . . . . . . 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 5288	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘ 𝐺) ∈ V)
17	16	ad2antrr 488	. . . . . . 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 6424	. . . . . . 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 1274	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})))
21		cnvco 4921	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
22	21	imaeq1i 5079	. . . . . . 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 5249	. . . . . . 7 ⊢ ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍})))
25		simprl 531	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → Fun 𝐹)
26	25	funfnd 5364	. . . . . . . . . 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 535	. . . . . . . . 9 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → 𝐹 ∈ 𝑉)
29		suppimacnvfn 6424	. . . . . . . . 9 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
30	27, 28, 18, 29	syl3anc 1274	. . . . . . . 8 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
31	30	imaeq2d 5082	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
32	24, 31	eqtr4id 2283	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
33	20, 23, 32	3eqtrd 2268	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
34	33	eleq2d 2301	. . . 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 713	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → (𝑥 ∈ ((𝐹 ∘ 𝐺) supp 𝑍) ↔ 𝑥 ∈ (◡𝐺 “ (𝐹 supp 𝑍))))
37	36	eqrdv 2229	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 1398   ∈ wcel 2202   ≠ wne 2403  {crab 2515  Vcvv 2803   ∖ cdif 3198  {csn 3673  ^◡ccnv 4730  dom cdm 4731   “ cima 4734   ∘ ccom 4735  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028   supp csupp 6413

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by:  supp0cosupp0fn  6445  imacosuppfn  6446