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Theorem imacosuppOLD 7865
 Description: Obsolete version of imacosupp 7864 as of 15-Sep-2023. The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
imacosuppOLD ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Proof of Theorem imacosuppOLD
StepHypRef Expression
1 cnvco 5721 . . . . . . . 8 (𝐹𝐺) = (𝐺𝐹)
21imaeq1i 5894 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
3 imaco 6074 . . . . . . 7 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
42, 3eqtri 2821 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
54imaeq2i 5895 . . . . 5 (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
6 funforn 6575 . . . . . . . 8 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
76biimpi 219 . . . . . . 7 (Fun 𝐺𝐺:dom 𝐺onto→ran 𝐺)
87ad2antrl 727 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → 𝐺:dom 𝐺onto→ran 𝐺)
9 simpl 486 . . . . . . . . . . . . 13 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
109anim2i 619 . . . . . . . . . . . 12 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
1110ancomd 465 . . . . . . . . . . 11 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
12 suppimacnv 7831 . . . . . . . . . . 11 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1311, 12syl 17 . . . . . . . . . 10 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1413sseq1d 3946 . . . . . . . . 9 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 ↔ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1514biimpd 232 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) ⊆ ran 𝐺 → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1615adantld 494 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺))
1716imp 410 . . . . . 6 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺)
18 foimacnv 6611 . . . . . 6 ((𝐺:dom 𝐺onto→ran 𝐺 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ ran 𝐺) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
198, 17, 18syl2anc 587 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ (𝐺 “ (𝐹 “ (V ∖ {𝑍})))) = (𝐹 “ (V ∖ {𝑍})))
205, 19syl5eq 2845 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
21 coexg 7623 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2221anim2i 619 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
2322ancomd 465 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
24 suppimacnv 7831 . . . . . . 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 5897 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2726adantr 484 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ((𝐹𝐺) “ (V ∖ {𝑍}))))
2813adantr 484 . . . 4 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2920, 27, 283eqtr4d 2843 . . 3 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
3029exp31 423 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐹 supp 𝑍))))
31 ima0 5913 . . . 4 (𝐺 “ ∅) = ∅
32 id 22 . . . . . . 7 𝑍 ∈ V → ¬ 𝑍 ∈ V)
3332intnand 492 . . . . . 6 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
34 supp0prc 7823 . . . . . 6 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
3533, 34syl 17 . . . . 5 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
3635imaeq2d 5897 . . . 4 𝑍 ∈ V → (𝐺 “ ((𝐹𝐺) supp 𝑍)) = (𝐺 “ ∅))
3732intnand 492 . . . . 5 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
38 supp0prc 7823 . . . . 5 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3937, 38syl 17 . . . 4 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
4031, 36, 393eqtr4a 2859 . . 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 185 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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  ◡ccnv 5519  dom cdm 5520  ran crn 5521   “ cima 5523   ∘ ccom 5524  Fun wfun 6321  –onto→wfo 6325  (class class class)co 7140   supp csupp 7820 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fo 6333  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-supp 7821 This theorem is referenced by: (None)
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