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Theorem cofunexg 7077
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 5592 . . 3 Rel (𝐴𝐵)
2 relssdmrn 5615 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 5345 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
5 dmexg 7044 . . . . 5 (𝐵𝐶 → dom 𝐵 ∈ V)
6 ssexg 4764 . . . . 5 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴𝐵) ∈ V)
74, 5, 6sylancr 694 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
87adantl 482 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
9 rnco 5600 . . . 4 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
10 rnexg 7045 . . . . . 6 (𝐵𝐶 → ran 𝐵 ∈ V)
11 resfunexg 6433 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
1210, 11sylan2 491 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13 rnexg 7045 . . . . 5 ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
1412, 13syl 17 . . . 4 ((Fun 𝐴𝐵𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
159, 14syl5eqel 2702 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
16 xpexg 6913 . . 3 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
178, 15, 16syl2anc 692 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
18 ssexg 4764 . 2 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
193, 17, 18sylancr 694 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186  wss 3555   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076  ccom 5078  Rel wrel 5079  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  cofunex2g  7078  fin1a2lem7  9172  revco  13517  ccatco  13518  lswco  13521  isofval  16338  bcthlem4  23032  sseqval  30228  sinccvglem  31271  pfxco  40734
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