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Theorem fpr2g 6475
Description: A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Assertion
Ref Expression
fpr2g |- ( ( A e. V /\ B e. W ) -> ( F : { A , B } --> C <-> ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) )
Proof of Theorem fpr2g
StepHypRef Expression
1 simpr 477 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> F : { A , B } --> C )
2 prid1g 4295 . . . . 5 |- ( A e. V -> A e. { A , B } )
32ad2antrr 762 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> A e. { A , B } )
41, 3ffvelrnd 6360 . . 3 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> ( F ` A ) e. C )
5 prid2g 4296 . . . . 5 |- ( B e. W -> B e. { A , B } )
65ad2antlr 763 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> B e. { A , B } )
71, 6ffvelrnd 6360 . . 3 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> ( F ` B ) e. C )
8 ffn 6045 . . . . 5 |- ( F : { A , B } --> C -> F Fn { A , B } )
98adantl 482 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> F Fn { A , B } )
10 fnpr2g 6474 . . . . 5 |- ( ( A e. V /\ B e. W ) -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
1110adantr 481 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
129, 11mpbid 222 . . 3 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
134, 7, 123jca 1242 . 2 |- ( ( ( A e. V /\ B e. W ) /\ F : { A , B } --> C ) -> ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
1410biimpar 502 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> F Fn { A , B } )
15143ad2antr3 1228 . . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> F Fn { A , B } )
16 simpr3 1069 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
172ad2antrr 762 . . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> A e. { A , B } )
18 simpr1 1067 . . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> ( F ` A ) e. C )
19 opelxpi 5148 . . . . . . 7 |- ( ( A e. { A , B } /\ ( F ` A ) e. C ) -> <. A , ( F ` A ) >. e. ( { A , B } X. C ) )
2017, 18, 19syl2anc 693 . . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> <. A , ( F ` A ) >. e. ( { A , B } X. C ) )
215ad2antlr 763 . . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> B e. { A , B } )
22 simpr2 1068 . . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> ( F ` B ) e. C )
23 opelxpi 5148 . . . . . . 7 |- ( ( B e. { A , B } /\ ( F ` B ) e. C ) -> <. B , ( F ` B ) >. e. ( { A , B } X. C ) )
2421, 22, 23syl2anc 693 . . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> <. B , ( F ` B ) >. e. ( { A , B } X. C ) )
2520, 24jca 554 . . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> ( <. A , ( F ` A ) >. e. ( { A , B } X. C ) /\ <. B , ( F ` B ) >. e. ( { A , B } X. C ) ) )
26 opex 4932 . . . . . 6 |- <. A , ( F ` A ) >. e. _V
27 opex 4932 . . . . . 6 |- <. B , ( F ` B ) >. e. _V
2826, 27prss 4351 . . . . 5 |- ( ( <. A , ( F ` A ) >. e. ( { A , B } X. C ) /\ <. B , ( F ` B ) >. e. ( { A , B } X. C ) ) <-> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } C_ ( { A , B } X. C ) )
2925, 28sylib 208 . . . 4 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } C_ ( { A , B } X. C ) )
3016, 29eqsstrd 3639 . . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> F C_ ( { A , B } X. C ) )
31 dff2 6371 . . 3 |- ( F : { A , B } --> C <-> ( F Fn { A , B } /\ F C_ ( { A , B } X. C ) ) )
3215, 30, 31sylanbrc 698 . 2 |- ( ( ( A e. V /\ B e. W ) /\ ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) -> F : { A , B } --> C )
3313, 32impbida 877 1 |- ( ( A e. V /\ B e. W ) -> ( F : { A , B } --> C <-> ( ( F ` A ) e. C /\ ( F ` B ) e. C /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {cpr 4179   <.cop 4183   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888
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
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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896
This theorem is referenced by:  f1prex  6539  uhgrwkspthlem2  26650
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