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Theorem imacosupp 7566
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 5509 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5673 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 5854 . . . . . . 7 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2828 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
54imaeq2i 5674 . . . . 5 (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
6 funforn 6334 . . . . . . . 8 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
76biimpi 207 . . . . . . 7 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
87ad2antrl 710 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺onto→ran 𝐺)
9 simpl 470 . . . . . . . . . . . . 13 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
109anim2i 605 . . . . . . . . . . . 12 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
1110ancomd 451 . . . . . . . . . . 11 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
12 suppimacnv 7536 . . . . . . . . . . 11 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1311, 12syl 17 . . . . . . . . . 10 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1413sseq1d 3829 . . . . . . . . 9 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1514biimpd 220 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1615adantld 480 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1716imp 395 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18 foimacnv 6366 . . . . . 6 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
198, 17, 18syl2anc 575 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
205, 19syl5eq 2852 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
21 coexg 7343 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2221anim2i 605 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
2322ancomd 451 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
24 suppimacnv 7536 . . . . . . 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 5676 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2726adantr 468 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2813adantr 468 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2920, 27, 283eqtr4d 2850 . . 3 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
3029exp31 408 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31 ima0 5691 . . . 4 (𝐺 “ ∅) = ∅
32 id 22 . . . . . . 7 𝑍 ∈ V → ¬ 𝑍 ∈ V)
3332intnand 478 . . . . . 6 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
34 supp0prc 7528 . . . . . 6 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
3533, 34syl 17 . . . . 5 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
3635imaeq2d 5676 . . . 4 𝑍 ∈ V → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ∅))
3732intnand 478 . . . . 5 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38 supp0prc 7528 . . . . 5 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3937, 38syl 17 . . . 4 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
4031, 36, 393eqtr4a 2866 . . 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 384   = wceq 1637  wcel 2156  Vcvv 3391  cdif 3766  wss 3769  c0 4116  {csn 4370  ccnv 5310  dom cdm 5311  ran crn 5312  cima 5314  ccom 5315  Fun wfun 6091  ontowfo 6095  (class class class)co 6870   supp csupp 7525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fo 6103  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-supp 7526
This theorem is referenced by:  gsumval3lem1  18503  gsumval3lem2  18504
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