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Theorem suppcoss 7949
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 7944, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.)
Hypotheses
Ref Expression
suppcoss.f (𝜑𝐹 Fn 𝐴)
suppcoss.g (𝜑𝐺:𝐵𝐴)
suppcoss.b (𝜑𝐵𝑊)
suppcoss.y (𝜑𝑌𝑉)
suppcoss.1 (𝜑 → (𝐹𝑌) = 𝑍)
Assertion
Ref Expression
suppcoss (𝜑 → ((𝐹𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Proof of Theorem suppcoss
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 suppcoss.f . . . 4 (𝜑𝐹 Fn 𝐴)
2 dffn3 6558 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 221 . . 3 (𝜑𝐹:𝐴⟶ran 𝐹)
4 suppcoss.g . . 3 (𝜑𝐺:𝐵𝐴)
53, 4fcod 6571 . 2 (𝜑 → (𝐹𝐺):𝐵⟶ran 𝐹)
6 eldif 3876 . . . . 5 (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
74ffnd 6546 . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
8 suppcoss.b . . . . . . . . 9 (𝜑𝐵𝑊)
9 suppcoss.y . . . . . . . . 9 (𝜑𝑌𝑉)
10 elsuppfn 7913 . . . . . . . . 9 ((𝐺 Fn 𝐵𝐵𝑊𝑌𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
117, 8, 9, 10syl3anc 1373 . . . . . . . 8 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1211notbid 321 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1312anbi2d 632 . . . . . 6 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌))))
14 annotanannot 835 . . . . . 6 ((𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌))
1513, 14bitrdi 290 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
166, 15syl5bb 286 . . . 4 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
17 nne 2944 . . . . . 6 (¬ (𝐺𝑘) ≠ 𝑌 ↔ (𝐺𝑘) = 𝑌)
1817anbi2i 626 . . . . 5 ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌))
194adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝐺:𝐵𝐴)
20 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝑘𝐵)
2119, 20fvco3d 6811 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = (𝐹‘(𝐺𝑘)))
22 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐺𝑘) = 𝑌)
2322fveq2d 6721 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹‘(𝐺𝑘)) = (𝐹𝑌))
24 suppcoss.1 . . . . . . . 8 (𝜑 → (𝐹𝑌) = 𝑍)
2524adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹𝑌) = 𝑍)
2621, 23, 253eqtrd 2781 . . . . . 6 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍)
2726ex 416 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ (𝐺𝑘) = 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2818, 27syl5bi 245 . . . 4 (𝜑 → ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2916, 28sylbid 243 . . 3 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍))
3029imp 410 . 2 ((𝜑𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹𝐺)‘𝑘) = 𝑍)
315, 30suppss 7936 1 (𝜑 → ((𝐹𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  cdif 3863  wss 3866  ran crn 5552  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213   supp csupp 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-supp 7904
This theorem is referenced by:  mplsubglem  20961  mhpinvcl  21092
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