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Theorem cofunexg 5864
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 4916 . . 3 Rel (𝐴𝐵)
2 relssdmrn 4938 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
31, 2ax-mp 7 . 2 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
4 dmcoss 4690 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
5 dmexg 4685 . . . . 5 (𝐵𝐶 → dom 𝐵 ∈ V)
6 ssexg 3970 . . . . 5 ((dom (𝐴𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴𝐵) ∈ V)
74, 5, 6sylancr 405 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
87adantl 271 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
9 rnco 4924 . . . 4 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
10 rnexg 4686 . . . . . 6 (𝐵𝐶 → ran 𝐵 ∈ V)
11 resfunexg 5500 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
1210, 11sylan2 280 . . . . 5 ((Fun 𝐴𝐵𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13 rnexg 4686 . . . . 5 ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
1412, 13syl 14 . . . 4 ((Fun 𝐴𝐵𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
159, 14syl5eqel 2174 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
16 xpexg 4540 . . 3 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
178, 15, 16syl2anc 403 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
18 ssexg 3970 . 2 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
193, 17, 18sylancr 405 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  Vcvv 2619  wss 2997   × cxp 4426  dom cdm 4428  ran crn 4429  cres 4430  ccom 4432  Rel wrel 4433  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  cofunex2g  5865
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