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

Theorem suppco 8141
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 8143. (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 7870 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2 simpl 484 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝑍 ∈ V)
3 suppimacnv 8109 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
41, 2, 3syl2an2 685 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
5 cnvco 5845 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
65imaeq1i 6014 . . . . 5 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍})))
8 imaco 6207 . . . . 5 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
9 simprl 770 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝐹𝑉)
10 suppimacnv 8109 . . . . . . 7 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
119, 2, 10syl2anc 585 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1211imaeq2d 6017 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
138, 12eqtr4id 2792 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 supp 𝑍)))
144, 7, 133eqtrd 2777 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
1514ex 414 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
16 prcnel 3470 . . . . . 6 𝑍 ∈ V → ¬ 𝑍 ∈ V)
1716intnand 490 . . . . 5 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
18 supp0prc 8099 . . . . 5 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
1917, 18syl 17 . . . 4 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
2016intnand 490 . . . . . . 7 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21 supp0prc 8099 . . . . . . 7 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
2220, 21syl 17 . . . . . 6 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
2322imaeq2d 6017 . . . . 5 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
24 ima0 6033 . . . . 5 (𝐺 “ ∅) = ∅
2523, 24eqtrdi 2789 . . . 4 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
2619, 25eqtr4d 2776 . . 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 397   = wceq 1542  wcel 2107  Vcvv 3447  cdif 3911  c0 4286  {csn 4590  ccnv 5636  cima 5640  ccom 5641  (class class class)co 7361   supp csupp 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8097
This theorem is referenced by:  supp0cosupp0  8143  imacosupp  8144
  Copyright terms: Public domain W3C validator