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Theorem cofunexg 7659
 Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cofunexg ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem cofunexg
StepHypRef Expression
1 relco 6078 . . 3 Rel (𝐴𝐵)
2 relssdmrn 6102 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5816 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
5 dmexg 7618 . . . . 5 (𝐵𝐶 → dom 𝐵 ∈ V)
6 ssexg 5196 . . . . 5 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴𝐵) ∈ V)
74, 5, 6sylancr 590 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
87adantl 485 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
9 rnco 6086 . . . 4 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
10 rnexg 7619 . . . . . 6 (𝐵𝐶 → ran 𝐵 ∈ V)
11 resfunexg 6974 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
1210, 11sylan2 595 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13 rnexg 7619 . . . . 5 ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
1412, 13syl 17 . . . 4 ((Fun 𝐴𝐵𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
159, 14eqeltrid 2856 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
168, 15xpexd 7477 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
17 ssexg 5196 . 2 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
183, 16, 17sylancr 590 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3860   × cxp 5525  dom cdm 5527  ran crn 5528   ↾ cres 5529   ∘ ccom 5531  Rel wrel 5532  Fun wfun 6333 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 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347 This theorem is referenced by:  cofunex2g  7660  fin1a2lem7  9871  revco  14248  ccatco  14249  pfxco  14252  lswco  14253  isofval  17091  bcthlem4  24032  sseqval  31878  sinccvglem  33150
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