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Theorem imacosupp 7380
Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
Assertion
Ref Expression
imacosupp ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Proof of Theorem imacosupp
StepHypRef Expression
1 cnvco 5340 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5498 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 5678 . . . . . . 7 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2673 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
54imaeq2i 5499 . . . . 5 (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
6 funforn 6160 . . . . . . . 8 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
76biimpi 206 . . . . . . 7 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
87ad2antrl 764 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺onto→ran 𝐺)
9 simpl 472 . . . . . . . . . . . . 13 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
109anim2i 592 . . . . . . . . . . . 12 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
1110ancomd 466 . . . . . . . . . . 11 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
12 suppimacnv 7351 . . . . . . . . . . 11 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1311, 12syl 17 . . . . . . . . . 10 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1413sseq1d 3665 . . . . . . . . 9 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1514biimpd 219 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1615adantld 482 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1716imp 444 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18 foimacnv 6192 . . . . . 6 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
198, 17, 18syl2anc 694 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
205, 19syl5eq 2697 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
21 coexg 7159 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2221anim2i 592 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
2322ancomd 466 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
24 suppimacnv 7351 . . . . . . 7 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
2523, 24syl 17 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
2625imaeq2d 5501 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2726adantr 480 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2813adantr 480 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2920, 27, 283eqtr4d 2695 . . 3 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
3029exp31 629 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31 ima0 5516 . . . 4 (𝐺 “ ∅) = ∅
32 id 22 . . . . . . 7 𝑍 ∈ V → ¬ 𝑍 ∈ V)
3332intnand 982 . . . . . 6 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
34 supp0prc 7343 . . . . . 6 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
3533, 34syl 17 . . . . 5 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
3635imaeq2d 5501 . . . 4 𝑍 ∈ V → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ∅))
3732intnand 982 . . . . 5 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38 supp0prc 7343 . . . . 5 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3937, 38syl 17 . . . 4 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
4031, 36, 393eqtr4a 2711 . . 3 𝑍 ∈ V → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
41402a1d 26 . 2 𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
4230, 41pm2.61i 176 1 ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Fun wfun 5920  ontowfo 5924  (class class class)co 6690   supp csupp 7340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341
This theorem is referenced by:  gsumval3lem1  18352  gsumval3lem2  18353
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