Theorem funcnvqpOLD 5953

Description: Obsolete proof of funcnvqp 5952 as of 14-Jul-2021. (Contributed by AV, 23-Jan-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
funcnvqpOLD	|- ( ( ( ( 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 funcnvqpOLD

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( A e. U /\ C e. V ) -> A e. U )
2	1	adantr 481	. . . 4 |- ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) -> A e. U )
3		simpr 477	. . . . 5 |- ( ( A e. U /\ C e. V ) -> C e. V )
4	3	adantr 481	. . . 4 |- ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) -> C e. V )
5		simp11 1091	. . . 4 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> B =/= D )
6		funcnvpr 5950	. . . 4 |- ( ( A e. U /\ C e. V /\ B =/= D ) -> Fun `' { <. A , B >. , <. C , D >. } )
7	2, 4, 5, 6	syl2an3an 1386	. . 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 >. } )
8		simpl 473	. . . . 5 |- ( ( E e. W /\ G e. T ) -> E e. W )
9	8	adantl 482	. . . 4 |- ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) -> E e. W )
10		simpr 477	. . . . 5 |- ( ( E e. W /\ G e. T ) -> G e. T )
11	10	adantl 482	. . . 4 |- ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) ) -> G e. T )
12		simp3 1063	. . . 4 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> F =/= H )
13		funcnvpr 5950	. . . 4 |- ( ( E e. W /\ G e. T /\ F =/= H ) -> Fun `' { <. E , F >. , <. G , H >. } )
14	9, 11, 12, 13	syl2an3an 1386	. . 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 >. } )
15		df-rn 5125	. . . . . 6 |- ran { <. A , B >. , <. C , D >. } = dom `' { <. A , B >. , <. C , D >. }
16		rnpropg 5615	. . . . . 6 |- ( ( A e. U /\ C e. V ) -> ran { <. A , B >. , <. C , D >. } = { B , D } )
17	15, 16	syl5eqr 2670	. . . . 5 |- ( ( A e. U /\ C e. V ) -> dom `' { <. A , B >. , <. C , D >. } = { B , D } )
18		df-rn 5125	. . . . . 6 |- ran { <. E , F >. , <. G , H >. } = dom `' { <. E , F >. , <. G , H >. }
19		rnpropg 5615	. . . . . 6 |- ( ( E e. W /\ G e. T ) -> ran { <. E , F >. , <. G , H >. } = { F , H } )
20	18, 19	syl5eqr 2670	. . . . 5 |- ( ( E e. W /\ G e. T ) -> dom `' { <. E , F >. , <. G , H >. } = { F , H } )
21	17, 20	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 } ) )
22		simp2 1062	. . . . . . 7 |- ( ( B =/= D /\ B =/= F /\ B =/= H ) -> B =/= F )
23		simpl 473	. . . . . . 7 |- ( ( D =/= F /\ D =/= H ) -> D =/= F )
24	22, 23	anim12i 590	. . . . . 6 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) ) -> ( B =/= F /\ D =/= F ) )
25	24	3adant3 1081	. . . . 5 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> ( B =/= F /\ D =/= F ) )
26		simp3 1063	. . . . . . 7 |- ( ( B =/= D /\ B =/= F /\ B =/= H ) -> B =/= H )
27		simpr 477	. . . . . . 7 |- ( ( D =/= F /\ D =/= H ) -> D =/= H )
28	26, 27	anim12i 590	. . . . . 6 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) ) -> ( B =/= H /\ D =/= H ) )
29	28	3adant3 1081	. . . . 5 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> ( B =/= H /\ D =/= H ) )
30		disjpr2 4248	. . . . 5 |- ( ( ( B =/= F /\ D =/= F ) /\ ( B =/= H /\ D =/= H ) ) -> ( { B , D } i^i { F , H } ) = (/) )
31	25, 29, 30	syl2anc 693	. . . 4 |- ( ( ( B =/= D /\ B =/= F /\ B =/= H ) /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) -> ( { B , D } i^i { F , H } ) = (/) )
32	21, 31	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 >. } ) = (/) )
33		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 >. } ) )
34	7, 14, 32, 33	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 >. } ) )
35		cnvun 5538	. . 3 |- `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) = ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } )
36	35	funeqi 5909	. 2 |- ( Fun `' ( { <. A , B >. , <. C , D >. } u. { <. E , F >. , <. G , H >. } ) <-> Fun ( `' { <. A , B >. , <. C , D >. } u. `' { <. E , F >. , <. G , H >. } ) )
37	34, 36	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: (None)