Theorem rtrclexlem 37923

Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)

Assertion

Ref	Expression
rtrclexlem	|- ( R e. V -> ( R u. ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) ) e. _V )

Proof of Theorem rtrclexlem

Step	Hyp	Ref	Expression
1		dmexg 7097	. . . 4 |- ( R e. V -> dom R e. _V )
2		rnexg 7098	. . . 4 |- ( R e. V -> ran R e. _V )
3		unexg 6959	. . . 4 |- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R u. ran R ) e. _V )
4	1, 2, 3	syl2anc 693	. . 3 |- ( R e. V -> ( dom R u. ran R ) e. _V )
5		sqxpexg 6963	. . 3 |- ( ( dom R u. ran R ) e. _V -> ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) e. _V )
6	4, 5	syl 17	. 2 |- ( R e. V -> ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) e. _V )
7		unexg 6959	. 2 |- ( ( R e. V /\ ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) e. _V ) -> ( R u. ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) ) e. _V )
8	6, 7	mpdan 702	1 |- ( R e. V -> ( R u. ( ( dom R u. ran R ) X. ( dom R u. ran R ) ) ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   u. cun 3572   X. cxp 5112   dom cdm 5114   ran crn 5115

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  rtrclex  37924