Theorem suppcoss 8186

Description: The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8181, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.)

Hypotheses

Ref	Expression
suppcoss.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
suppcoss.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
suppcoss.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
suppcoss.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
suppcoss.1	⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)

Assertion

Ref	Expression
suppcoss	⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Proof of Theorem suppcoss

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppcoss.f	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		dffn3 6700	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
4		suppcoss.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
5	3, 4	fcod 6713	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ran 𝐹)
6		eldif 3924	. . . . 5 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
7	4	ffnd 6689	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
8		suppcoss.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		suppcoss.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		elsuppfn 8149	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑌 ∈ 𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
11	7, 8, 9, 10	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
12	11	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌))))
14		annotanannot 834	. . . . . 6 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) ≠ 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌))
15	13, 14	bitrdi 287	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
16	6, 15	bitrid 283	. . . 4 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌)))
17		nne 2929	. . . . . 6 ⊢ (¬ (𝐺‘𝑘) ≠ 𝑌 ↔ (𝐺‘𝑘) = 𝑌)
18	17	anbi2i 623	. . . . 5 ⊢ ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) ↔ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌))
19	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝐺:𝐵⟶𝐴)
20		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → 𝑘 ∈ 𝐵)
21	19, 20	fvco3d 6961	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = (𝐹‘(𝐺‘𝑘)))
22		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐺‘𝑘) = 𝑌)
23	22	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘𝑌))
24		suppcoss.1	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → (𝐹‘𝑌) = 𝑍)
26	21, 23, 25	3eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
27	26	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ (𝐺‘𝑘) = 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
28	18, 27	biimtrid 242	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ∧ ¬ (𝐺‘𝑘) ≠ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
29	16, 28	sylbid 240	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍))
30	29	imp 406	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹 ∘ 𝐺)‘𝑘) = 𝑍)
31	5, 30	suppss 8173	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911   ⊆ wss 3914  ran crn 5639   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   supp csupp 8139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-supp 8140

This theorem is referenced by:  mplsubglem  21908  mhpinvcl  22039