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Theorem imacosupp 8144
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 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺) β†’ (𝐺 β€œ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))

Proof of Theorem imacosupp
StepHypRef Expression
1 suppco 8141 . . . 4 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ ((𝐹 ∘ 𝐺) supp 𝑍) = (◑𝐺 β€œ (𝐹 supp 𝑍)))
21imaeq2d 6017 . . 3 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ (𝐺 β€œ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 β€œ (◑𝐺 β€œ (𝐹 supp 𝑍))))
3 funforn 6767 . . . 4 (Fun 𝐺 ↔ 𝐺:dom 𝐺–ontoβ†’ran 𝐺)
4 foimacnv 6805 . . . 4 ((𝐺:dom 𝐺–ontoβ†’ran 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺) β†’ (𝐺 β€œ (◑𝐺 β€œ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
53, 4sylanb 582 . . 3 ((Fun 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺) β†’ (𝐺 β€œ (◑𝐺 β€œ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
62, 5sylan9eq 2793 . 2 (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺)) β†’ (𝐺 β€œ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))
76ex 414 1 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺) β†’ (𝐺 β€œ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3914  β—‘ccnv 5636  dom cdm 5637  ran crn 5638   β€œ cima 5640   ∘ ccom 5641  Fun wfun 6494  β€“ontoβ†’wfo 6498  (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-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-supp 8097
This theorem is referenced by:  gsumval3lem1  19690  gsumval3lem2  19691
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