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Theorem imacosupp 7222
 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 5230 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5382 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 5557 . . . . . . 7 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2632 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
54imaeq2i 5383 . . . . 5 (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
6 funforn 6035 . . . . . . . 8 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
76biimpi 205 . . . . . . 7 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
87ad2antrl 760 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺onto→ran 𝐺)
9 simpl 472 . . . . . . . . . . . . 13 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
109anim2i 591 . . . . . . . . . . . 12 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
1110ancomd 466 . . . . . . . . . . 11 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
12 suppimacnv 7193 . . . . . . . . . . 11 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1311, 12syl 17 . . . . . . . . . 10 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1413sseq1d 3595 . . . . . . . . 9 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1514biimpd 218 . . . . . . . 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 6067 . . . . . 6 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
198, 17, 18syl2anc 691 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
205, 19syl5eq 2656 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
21 coexg 7010 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2221anim2i 591 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
2322ancomd 466 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
24 suppimacnv 7193 . . . . . . 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 5385 . . . . 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 2654 . . 3 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
3029exp31 628 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31 ima0 5400 . . . 4 (𝐺 “ ∅) = ∅
32 id 22 . . . . . . 7 𝑍 ∈ V → ¬ 𝑍 ∈ V)
3332intnand 953 . . . . . 6 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
34 supp0prc 7185 . . . . . 6 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
3533, 34syl 17 . . . . 5 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
3635imaeq2d 5385 . . . 4 𝑍 ∈ V → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ∅))
3732intnand 953 . . . . 5 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38 supp0prc 7185 . . . . 5 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3937, 38syl 17 . . . 4 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
4031, 36, 393eqtr4a 2670 . . 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 175 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 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  –onto→wfo 5802  (class class class)co 6549   supp csupp 7182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183 This theorem is referenced by:  gsumval3lem1  18129  gsumval3lem2  18130
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