Theorem cofunexg 7898

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 6067	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relssdmrn 6227	. . 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 5924	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5		dmexg 7848	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → dom 𝐵 ∈ V)
6		ssexg 5258	. . . . 5 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ dom 𝐵 ∈ V) → dom (𝐴 ∘ 𝐵) ∈ V)
7	4, 5, 6	sylancr 593	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ∘ 𝐵) ∈ V)
8	7	adantl 482	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ∘ 𝐵) ∈ V)
9		rnco 6210	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
10		rnexg 7849	. . . . . 6 ⊢ (𝐵 ∈ 𝐶 → ran 𝐵 ∈ V)
11		resfunexg 7166	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐵 ∈ V) → (𝐴 ↾ ran 𝐵) ∈ V)
12	10, 11	sylan2 599	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ ran 𝐵) ∈ V)
13		rnexg 7849	. . . . 5 ⊢ ((𝐴 ↾ ran 𝐵) ∈ V → ran (𝐴 ↾ ran 𝐵) ∈ V)
14	12, 13	syl 17	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ ran 𝐵) ∈ V)
15	9, 14	eqeltrid 2844	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ∘ 𝐵) ∈ V)
16	8, 15	xpexd 7701	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V)
17		ssexg 5258	. 2 ⊢ (((𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∧ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
18	3, 16, 17	sylancr 593	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890   × cxp 5623  dom cdm 5625  ran crn 5626   ↾ cres 5627   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  cofunex2g  7899  fin1a2lem7  10326  revco  14794  ccatco  14795  pfxco  14798  lswco  14799  isofval  17722  bcthlem4  25319  sseqval  34579  sinccvglem  35907