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Theorem cofunexg 7513
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 5979 . . 3 Rel (𝐴𝐵)
2 relssdmrn 6003 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5730 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
5 dmexg 7476 . . . . 5 (𝐵𝐶 → dom 𝐵 ∈ V)
6 ssexg 5125 . . . . 5 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴𝐵) ∈ V)
74, 5, 6sylancr 587 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
87adantl 482 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
9 rnco 5987 . . . 4 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
10 rnexg 7477 . . . . . 6 (𝐵𝐶 → ran 𝐵 ∈ V)
11 resfunexg 6851 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
1210, 11sylan2 592 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13 rnexg 7477 . . . . 5 ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
1412, 13syl 17 . . . 4 ((Fun 𝐴𝐵𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
159, 14syl5eqel 2889 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
168, 15xpexd 7338 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
17 ssexg 5125 . 2 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
183, 16, 17sylancr 587 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2083  Vcvv 3440  wss 3865   × cxp 5448  dom cdm 5450  ran crn 5451  cres 5452  ccom 5454  Rel wrel 5455  Fun wfun 6226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240
This theorem is referenced by:  cofunex2g  7514  fin1a2lem7  9681  revco  14036  ccatco  14037  pfxco  14040  lswco  14041  isofval  16860  bcthlem4  23617  sseqval  31259  sinccvglem  32525
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