Theorem suppcoss 8149

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 8144, 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 6674	. . . 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 6687	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐵⟶ran 𝐹)
6		eldif 3911	. . . . 5 ⊢ (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
7	4	ffnd 6663	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
8		suppcoss.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		suppcoss.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
10		elsuppfn 8112	. . . . . . . . 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 2936	. . . . . 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 6934	. . . . . . 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 6838	. . . . . . 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 2775	. . . . . 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 8136	1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ⊆ wss 3901  ran crn 5625   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   supp csupp 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8103

This theorem is referenced by:  mplsubglem  21954  mhpinvcl  22095