Theorem suppcofn 6465

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 6435	. . . . 5 ⊢ supp = (𝑓 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑓 ∣ (𝑓 “ {𝑖}) ≠ {𝑧}})
2	1	elmpocl2 6250	. . . 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 5382	. . . . . . 7 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) → 𝐺 Fn dom 𝐺)
6		elpreima 5796	. . . . . . 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 6250	. . . . 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 5391	. . . . . . . . . 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 5382	. . . . . . . 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 5306	. . . . . . . 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 6445	. . . . . . 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 4939	. . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹)
22	21	imaeq1i 5097	. . . . . . 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 5267	. . . . . . 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 5382	. . . . . . . . . 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 6445	. . . . . . . . 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 5100	. . . . . . 7 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ (◡𝐹 “ (V ∖ {𝑍}))))
32	24, 31	eqtr4id 2284	. . . . . 6 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((◡𝐺 ∘ ◡𝐹) “ (V ∖ {𝑍})) = (◡𝐺 “ (𝐹 supp 𝑍)))
33	20, 23, 32	3eqtrd 2269	. . . . 5 ⊢ ((((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐹 ∧ Fun 𝐺)) ∧ 𝑍 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
34	33	eleq2d 2302	. . . 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 2230	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 2203   ≠ wne 2412  {crab 2524  Vcvv 2812   ∖ cdif 3207  {csn 3688  ^◡ccnv 4747  dom cdm 4748   “ cima 4751   ∘ ccom 4752  Fun wfun 5345   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  supp0cosupp0fn  6466  imacosuppfn  6467  fsuppcorn  7253