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Theorem suppcoss 7869
 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 7864, 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 6504 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 221 . . 3 (𝜑𝐹:𝐴⟶ran 𝐹)
4 suppcoss.g . . 3 (𝜑𝐺:𝐵𝐴)
53, 4fcod 6511 . 2 (𝜑 → (𝐹𝐺):𝐵⟶ran 𝐹)
6 eldif 3892 . . . . 5 (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
74ffnd 6493 . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
8 suppcoss.b . . . . . . . . 9 (𝜑𝐵𝑊)
9 suppcoss.y . . . . . . . . 9 (𝜑𝑌𝑉)
10 elsuppfn 7833 . . . . . . . . 9 ((𝐺 Fn 𝐵𝐵𝑊𝑌𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
117, 8, 9, 10syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1211notbid 321 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1312anbi2d 631 . . . . . 6 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌))))
14 annotanannot 833 . . . . . 6 ((𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌))
1513, 14syl6bb 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 2991 . . . . . 6 (¬ (𝐺𝑘) ≠ 𝑌 ↔ (𝐺𝑘) = 𝑌)
1817anbi2i 625 . . . . 5 ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌))
194adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝐺:𝐵𝐴)
20 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝑘𝐵)
2119, 20fvco3d 6745 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = (𝐹‘(𝐺𝑘)))
22 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐺𝑘) = 𝑌)
2322fveq2d 6656 . . . . . . 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 2837 . . . . . 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 7856 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 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3879   ⊆ wss 3882  ran crn 5523   ∘ ccom 5526   Fn wfn 6324  ⟶wf 6325  ‘cfv 6329  (class class class)co 7142   supp csupp 7823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-supp 7824 This theorem is referenced by:  mplsubglem  20707  mhpinvcl  20838
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