Theorem cofunexg 7950

Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cofunexg	⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)

Proof of Theorem cofunexg

Step	Hyp	Ref	Expression
1		relco 6108	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 6270	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))
4		dmcoss 5963	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		dmexg 7902	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → dom 𝐵 ∈ V)
6		ssexg 5288	. . . . 5 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴 ∘ 𝐵) ∈ V)
7	4, 5, 6	sylancr 599	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ∘ 𝐵) ∈ V)
8	7	adantl 487	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ∘ 𝐵) ∈ V)
9		rnco 6252	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
10		rnexg 7903	. . . . . 6 ⊢ (𝐵 ∈ 𝐶 → ran 𝐵 ∈ V)
11		resfunexg 7218	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
12	10, 11	sylan2 605	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13		rnexg 7903	. . . . 5 ⊢ ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
14	12, 13	syl 18	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
15	9, 14	eqeltrid 2866	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ∘ 𝐵) ∈ V)
16	8, 15	xpexd 7754	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V)
17		ssexg 5288	. 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∧ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
18	3, 16, 17	sylancr 599	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   × cxp 5657  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663  Rel wrel 5664  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  cofunex2g  7951  fin1a2lem7  10412  revco  14909  ccatco  14910  pfxco  14913  lswco  14914  isofval  17852  bcthlem4  25561  sseqval  34907  sinccvglem  36259