Theorem funcnvqp 5952

Description: The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
funcnvqp	|- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> Fun `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) )

Proof of Theorem funcnvqp

Step	Hyp	Ref	Expression
1		funcnvpr 5950	. . . . . . 7 |- ( ( A e. U /\ C e. V /\ B =/= D ) -> Fun `' { <. A , B >. , <. C , D >. } )
2	1	3expa 1265	. . . . . 6 |- ( ( ( A e. U /\ C e. V ) /\ B =/= D ) -> Fun `' { <. A , B >. , <. C , D >. } )
3	2	3ad2antr1 1226	. . . . 5 |- ( ( ( A e. U /\ C e. V ) /\ ( B =/= D /\ B =/= F /\ B =/= H ) ) -> Fun `' { <. A , B >. , <. C , D >. } )
4	3	ad2ant2r 783	. . . 4 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ F =/= H ) ) -> Fun `' { <. A , B >. , <. C , D >. } )
5	4	3adantr2 1221	. . 3 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> Fun `' { <. A , B >. , <. C , D >. } )
6		funcnvpr 5950	. . . . . 6 |- ( ( E e. W /\ G e. T /\ F =/= H ) -> Fun `' { <. E , F >. , <. G , H >. } )
7	6	3expa 1265	. . . . 5 |- ( ( ( E e. W /\ G e. T ) /\ F =/= H ) -> Fun `' { <. E , F >. , <. G , H >. } )
8	7	ad2ant2l 782	. . . 4 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ F =/= H ) ) -> Fun `' { <. E , F >. , <. G , H >. } )
9	8	3adantr2 1221	. . 3 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> Fun `' { <. E , F >. , <. G , H >. } )
10		df-rn 5125	. . . . . 6 |- ran { <. A , B >. , <. C , D >. } = dom `' { <. A , B >. , <. C , D >. }
11		rnpropg 5615	. . . . . 6 |- ( ( A e. U /\ C e. V ) -> ran { <. A , B >. , <. C , D >. } = { B , D } )
12	10, 11	syl5eqr 2670	. . . . 5 |- ( ( A e. U /\ C e. V ) -> dom `' { <. A , B >. , <. C , D >. } = { B , D } )
13		df-rn 5125	. . . . . 6 |- ran { <. E , F >. , <. G , H >. } = dom `' { <. E , F >. , <. G , H >. }
14		rnpropg 5615	. . . . . 6 |- ( ( E e. W /\ G e. T ) -> ran { <. E , F >. , <. G , H >. } = { F , H } )
15	13, 14	syl5eqr 2670	. . . . 5 |- ( ( E e. W /\ G e. T ) -> dom `' { <. E , F >. , <. G , H >. } = { F , H } )
16	12, 15	ineqan12d 3816	. . . 4 |- ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) -> ( dom `' { <. A , B >. , <. C , D >. } i^i dom `' { <. E , F >. , <. G , H >. } ) = ( { B , D } i^i { F , H } ) )
17		disjpr2 4248	. . . . . . 7 |- ( ( ( B =/= F /\ D =/= F ) /\ ( B =/= H /\ D =/= H ) ) -> ( { B , D } i^i { F , H } ) = (/) )
18	17	an4s 869	. . . . . 6 |- ( ( ( B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) ) -> ( { B , D } i^i { F , H } ) = (/) )
19	18	3adantl1 1217	. . . . 5 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) ) -> ( { B , D } i^i { F , H } ) = (/) )
20	19	3adant3 1081	. . . 4 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> ( { B , D } i^i { F , H } ) = (/) )
21	16, 20	sylan9eq 2676	. . 3 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> ( dom `' { <. A , B >. , <. C , D >. } i^i dom `' { <. E , F >. , <. G , H >. } ) = (/) )
22		funun 5932	. . 3 |- ( ( ( Fun `' { <. A , B >. , <. C , D >. } /\ Fun `' { <. E , F >. , <. G , H >. } ) /\ ( dom `' { <. A , B >. , <. C , D >. } i^i dom `' { <. E , F >. , <. G , H >. } ) = (/) ) -> Fun ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } ) )
23	5, 9, 21, 22	syl21anc 1325	. 2 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> Fun ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } ) )
24		cnvun 5538	. . 3 |- `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) = ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } )
25	24	funeqi 5909	. 2 |- ( Fun `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) <-> Fun ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } ) )
26	23, 25	sylibr 224	1 |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) /\ ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) ) -> Fun `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {cpr 4179   <.cop 4183   `'ccnv 5113   dom cdm 5114   ran crn 5115   Fun wfun 5882

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890

This theorem is referenced by:  funcnvs4  13660