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Theorem cofunexg 7934
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 6101 . . 3 Rel (𝐴𝐵)
2 relssdmrn 6261 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5964 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
5 dmexg 7891 . . . . 5 (𝐵𝐶 → dom 𝐵 ∈ V)
6 ssexg 5316 . . . . 5 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴𝐵) ∈ V)
74, 5, 6sylancr 586 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
87adantl 481 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
9 rnco 6245 . . . 4 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
10 rnexg 7892 . . . . . 6 (𝐵𝐶 → ran 𝐵 ∈ V)
11 resfunexg 7212 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
1210, 11sylan2 592 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13 rnexg 7892 . . . . 5 ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
1412, 13syl 17 . . . 4 ((Fun 𝐴𝐵𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
159, 14eqeltrid 2831 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
168, 15xpexd 7735 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
17 ssexg 5316 . 2 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
183, 16, 17sylancr 586 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3468  wss 3943   × cxp 5667  dom cdm 5669  ran crn 5670  cres 5671  ccom 5673  Rel wrel 5674  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  cofunex2g  7935  fin1a2lem7  10403  revco  14791  ccatco  14792  pfxco  14795  lswco  14796  isofval  17713  bcthlem4  25210  sseqval  33917  sinccvglem  35185
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