Theorem cofunexg 6342

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 5289	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 5311	. . 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 5054	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		dmexg 5048	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → dom 𝐵 ∈ V)
6		ssexg 4274	. . . . 5 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴 ∘ 𝐵) ∈ V)
7	4, 5, 6	sylancr 418	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ∘ 𝐵) ∈ V)
8	7	adantl 277	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ∘ 𝐵) ∈ V)
9		rnco 5297	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
10		rnexg 5049	. . . . . 6 ⊢ (𝐵 ∈ 𝐶 → ran 𝐵 ∈ V)
11		resfunexg 5939	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
12	10, 11	sylan2 286	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13		rnexg 5049	. . . . 5 ⊢ ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
14	12, 13	syl 14	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
15	9, 14	eqeltrid 2325	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ∘ 𝐵) ∈ V)
16		xpexg 4891	. . 3 ⊢ ((dom (𝐴 ∘ 𝐵) ∈ V ∧ ran (𝐴 ∘ 𝐵) ∈ V) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V)
17	8, 15, 16	syl2anc 415	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V)
18		ssexg 4274	. 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∧ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
19	3, 17, 18	sylancr 418	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   × cxp 4774  dom cdm 4776  ran crn 4777   ↾ cres 4778   ∘ ccom 4780  Rel wrel 4781  Fun wfun 5374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388

This theorem is used by:  cofunex2g  6343  ctm  7456  ctssdclemr  7459  prdsex  14280  prdsval  14281  prdsbaslemss  14282