Theorem wunco 9555

Description: A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	|- ( ph -> U e. WUni )
wunop.2	|- ( ph -> A e. U )
wunco.3	|- ( ph -> B e. U )

Assertion

Ref	Expression
wunco	|- ( ph -> ( A o. B ) e. U )

Proof of Theorem wunco

Step	Hyp	Ref	Expression
1		wun0.1	. 2 |- ( ph -> U e. WUni )
2		wunco.3	. . . . 5 |- ( ph -> B e. U )
3	1, 2	wundm 9550	. . . 4 |- ( ph -> dom B e. U )
4		dmcoss 5385	. . . . 5 |- dom ( A o. B ) C_ dom B
5	4	a1i 11	. . . 4 |- ( ph -> dom ( A o. B ) C_ dom B )
6	1, 3, 5	wunss 9534	. . 3 |- ( ph -> dom ( A o. B ) e. U )
7		wunop.2	. . . . 5 |- ( ph -> A e. U )
8	1, 7	wunrn 9551	. . . 4 |- ( ph -> ran A e. U )
9		rncoss 5386	. . . . 5 |- ran ( A o. B ) C_ ran A
10	9	a1i 11	. . . 4 |- ( ph -> ran ( A o. B ) C_ ran A )
11	1, 8, 10	wunss 9534	. . 3 |- ( ph -> ran ( A o. B ) e. U )
12	1, 6, 11	wunxp 9546	. 2 |- ( ph -> ( dom ( A o. B ) X. ran ( A o. B ) ) e. U )
13		relco 5633	. . 3 |- Rel ( A o. B )
14		relssdmrn 5656	. . 3 |- ( Rel ( A o. B ) -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
15	13, 14	mp1i 13	. 2 |- ( ph -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
16	1, 12, 15	wunss 9534	1 |- ( ph -> ( A o. B ) e. U )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   Rel wrel 5119  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-wun 9524

This theorem is referenced by: (None)