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Theorem imacosupp 8193
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 8190 . . . 4 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ ((𝐹 ∘ 𝐺) supp 𝑍) = (◑𝐺 β€œ (𝐹 supp 𝑍)))
21imaeq2d 6059 . . 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 6812 . . . 4 (Fun 𝐺 ↔ 𝐺:dom 𝐺–ontoβ†’ran 𝐺)
4 foimacnv 6850 . . . 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 581 . . 3 ((Fun 𝐺 ∧ (𝐹 supp 𝑍) βŠ† ran 𝐺) β†’ (𝐺 β€œ (◑𝐺 β€œ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍))
62, 5sylan9eq 2792 . 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 413 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 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  Fun wfun 6537  β€“ontoβ†’wfo 6541  (class class class)co 7408   supp csupp 8145
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-supp 8146
This theorem is referenced by:  gsumval3lem1  19772  gsumval3lem2  19773
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