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Theorem fnct 9359
Description: If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
fnct |- ( ( F Fn A /\ A ~<_ om ) -> F ~<_ om )
Proof of Theorem fnct
StepHypRef Expression
1 ctex 7970 . . . . 5 |- ( A ~<_ om -> A e. _V )
21adantl 482 . . . 4 |- ( ( F Fn A /\ A ~<_ om ) -> A e. _V )
3 fndm 5990 . . . . . . . 8 |- ( F Fn A -> dom F = A )
43eleq1d 2686 . . . . . . 7 |- ( F Fn A -> ( dom F e. _V <-> A e. _V ) )
54adantr 481 . . . . . 6 |- ( ( F Fn A /\ A ~<_ om ) -> ( dom F e. _V <-> A e. _V ) )
62, 5mpbird 247 . . . . 5 |- ( ( F Fn A /\ A ~<_ om ) -> dom F e. _V )
7 fnfun 5988 . . . . . 6 |- ( F Fn A -> Fun F )
87adantr 481 . . . . 5 |- ( ( F Fn A /\ A ~<_ om ) -> Fun F )
9 funrnex 7133 . . . . 5 |- ( dom F e. _V -> ( Fun F -> ran F e. _V ) )
106, 8, 9sylc 65 . . . 4 |- ( ( F Fn A /\ A ~<_ om ) -> ran F e. _V )
11 xpexg 6960 . . . 4 |- ( ( A e. _V /\ ran F e. _V ) -> ( A X. ran F ) e. _V )
122, 10, 11syl2anc 693 . . 3 |- ( ( F Fn A /\ A ~<_ om ) -> ( A X. ran F ) e. _V )
13 simpl 473 . . . . 5 |- ( ( F Fn A /\ A ~<_ om ) -> F Fn A )
14 dffn3 6054 . . . . 5 |- ( F Fn A <-> F : A --> ran F )
1513, 14sylib 208 . . . 4 |- ( ( F Fn A /\ A ~<_ om ) -> F : A --> ran F )
16 fssxp 6060 . . . 4 |- ( F : A --> ran F -> F C_ ( A X. ran F ) )
1715, 16syl 17 . . 3 |- ( ( F Fn A /\ A ~<_ om ) -> F C_ ( A X. ran F ) )
18 ssdomg 8001 . . 3 |- ( ( A X. ran F ) e. _V -> ( F C_ ( A X. ran F ) -> F ~<_ ( A X. ran F ) ) )
1912, 17, 18sylc 65 . 2 |- ( ( F Fn A /\ A ~<_ om ) -> F ~<_ ( A X. ran F ) )
20 xpdom1g 8057 . . . . 5 |- ( ( ran F e. _V /\ A ~<_ om ) -> ( A X. ran F ) ~<_ ( om X. ran F ) )
2110, 20sylancom 701 . . . 4 |- ( ( F Fn A /\ A ~<_ om ) -> ( A X. ran F ) ~<_ ( om X. ran F ) )
22 omex 8540 . . . . 5 |- om e. _V
23 fnrndomg 9358 . . . . . . 7 |- ( A e. _V -> ( F Fn A -> ran F ~<_ A ) )
242, 13, 23sylc 65 . . . . . 6 |- ( ( F Fn A /\ A ~<_ om ) -> ran F ~<_ A )
25 domtr 8009 . . . . . 6 |- ( ( ran F ~<_ A /\ A ~<_ om ) -> ran F ~<_ om )
2624, 25sylancom 701 . . . . 5 |- ( ( F Fn A /\ A ~<_ om ) -> ran F ~<_ om )
27 xpdom2g 8056 . . . . 5 |- ( ( om e. _V /\ ran F ~<_ om ) -> ( om X. ran F ) ~<_ ( om X. om ) )
2822, 26, 27sylancr 695 . . . 4 |- ( ( F Fn A /\ A ~<_ om ) -> ( om X. ran F ) ~<_ ( om X. om ) )
29 domtr 8009 . . . 4 |- ( ( ( A X. ran F ) ~<_ ( om X. ran F ) /\ ( om X. ran F ) ~<_ ( om X. om ) ) -> ( A X. ran F ) ~<_ ( om X. om ) )
3021, 28, 29syl2anc 693 . . 3 |- ( ( F Fn A /\ A ~<_ om ) -> ( A X. ran F ) ~<_ ( om X. om ) )
31 xpomen 8838 . . 3 |- ( om X. om ) ~~ om
32 domentr 8015 . . 3 |- ( ( ( A X. ran F ) ~<_ ( om X. om ) /\ ( om X. om ) ~~ om ) -> ( A X. ran F ) ~<_ om )
3330, 31, 32sylancl 694 . 2 |- ( ( F Fn A /\ A ~<_ om ) -> ( A X. ran F ) ~<_ om )
34 domtr 8009 . 2 |- ( ( F ~<_ ( A X. ran F ) /\ ( A X. ran F ) ~<_ om ) -> F ~<_ om )
3519, 33, 34syl2anc 693 1 |- ( ( F Fn A /\ A ~<_ om ) -> F ~<_ om )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   omcom 7065   ~~ cen 7952   ~<_ cdom 7953
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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939
This theorem is referenced by:  mptct  9360  mpt2cti  29493  mptctf  29495  omssubadd  30362
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