MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppco Structured version   Visualization version   GIF version

Theorem suppco 8137
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.) Extract this statement from the proof of supp0cosupp0 8139. (Revised by SN, 15-Sep-2023.)
Assertion
Ref Expression
suppco ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))

Proof of Theorem suppco
StepHypRef Expression
1 coexg 7866 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2 simpl 483 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝑍 ∈ V)
3 suppimacnv 8105 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
41, 2, 3syl2an2 684 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
5 cnvco 5841 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
65imaeq1i 6010 . . . . 5 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍})))
8 imaco 6203 . . . . 5 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
9 simprl 769 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝐹𝑉)
10 suppimacnv 8105 . . . . . . 7 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
119, 2, 10syl2anc 584 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1211imaeq2d 6013 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
138, 12eqtr4id 2795 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 supp 𝑍)))
144, 7, 133eqtrd 2780 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
1514ex 413 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
16 prcnel 3468 . . . . . 6 𝑍 ∈ V → ¬ 𝑍 ∈ V)
1716intnand 489 . . . . 5 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
18 supp0prc 8095 . . . . 5 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
1917, 18syl 17 . . . 4 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
2016intnand 489 . . . . . . 7 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21 supp0prc 8095 . . . . . . 7 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
2220, 21syl 17 . . . . . 6 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
2322imaeq2d 6013 . . . . 5 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
24 ima0 6029 . . . . 5 (𝐺 “ ∅) = ∅
2523, 24eqtrdi 2792 . . . 4 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
2619, 25eqtr4d 2779 . . 3 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
2726a1d 25 . 2 𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
2815, 27pm2.61i 182 1 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  c0 4282  {csn 4586  ccnv 5632  cima 5636  ccom 5637  (class class class)co 7357   supp csupp 8092
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-supp 8093
This theorem is referenced by:  supp0cosupp0  8139  imacosupp  8140
  Copyright terms: Public domain W3C validator