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Theorem fsuppcurry2 31690
Description: Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
Hypotheses
Ref Expression
fsuppcurry2.g šŗ = (š‘„ āˆˆ š“ ā†¦ (š‘„š¹š¶))
fsuppcurry2.z (šœ‘ ā†’ š‘ āˆˆ š‘ˆ)
fsuppcurry2.a (šœ‘ ā†’ š“ āˆˆ š‘‰)
fsuppcurry2.b (šœ‘ ā†’ šµ āˆˆ š‘Š)
fsuppcurry2.f (šœ‘ ā†’ š¹ Fn (š“ Ɨ šµ))
fsuppcurry2.c (šœ‘ ā†’ š¶ āˆˆ šµ)
fsuppcurry2.1 (šœ‘ ā†’ š¹ finSupp š‘)
Assertion
Ref Expression
fsuppcurry2 (šœ‘ ā†’ šŗ finSupp š‘)
Distinct variable groups:   š‘„,š“   š‘„,š¶   š‘„,š¹
Allowed substitution hints:   šœ‘(š‘„)   šµ(š‘„)   š‘ˆ(š‘„)   šŗ(š‘„)   š‘‰(š‘„)   š‘Š(š‘„)   š‘(š‘„)

Proof of Theorem fsuppcurry2
Dummy variables š‘¦ š‘§ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsuppcurry2.g . . . 4 šŗ = (š‘„ āˆˆ š“ ā†¦ (š‘„š¹š¶))
2 oveq1 7365 . . . . 5 (š‘„ = š‘¦ ā†’ (š‘„š¹š¶) = (š‘¦š¹š¶))
32cbvmptv 5219 . . . 4 (š‘„ āˆˆ š“ ā†¦ (š‘„š¹š¶)) = (š‘¦ āˆˆ š“ ā†¦ (š‘¦š¹š¶))
41, 3eqtri 2761 . . 3 šŗ = (š‘¦ āˆˆ š“ ā†¦ (š‘¦š¹š¶))
5 fsuppcurry2.a . . . 4 (šœ‘ ā†’ š“ āˆˆ š‘‰)
65mptexd 7175 . . 3 (šœ‘ ā†’ (š‘¦ āˆˆ š“ ā†¦ (š‘¦š¹š¶)) āˆˆ V)
74, 6eqeltrid 2838 . 2 (šœ‘ ā†’ šŗ āˆˆ V)
81funmpt2 6541 . . 3 Fun šŗ
98a1i 11 . 2 (šœ‘ ā†’ Fun šŗ)
10 fsuppcurry2.z . 2 (šœ‘ ā†’ š‘ āˆˆ š‘ˆ)
11 fo1st 7942 . . . . 5 1st :Vā€“ontoā†’V
12 fofun 6758 . . . . 5 (1st :Vā€“ontoā†’V ā†’ Fun 1st )
1311, 12ax-mp 5 . . . 4 Fun 1st
14 funres 6544 . . . 4 (Fun 1st ā†’ Fun (1st ā†¾ (V Ɨ V)))
1513, 14mp1i 13 . . 3 (šœ‘ ā†’ Fun (1st ā†¾ (V Ɨ V)))
16 fsuppcurry2.1 . . . 4 (šœ‘ ā†’ š¹ finSupp š‘)
1716fsuppimpd 9316 . . 3 (šœ‘ ā†’ (š¹ supp š‘) āˆˆ Fin)
18 imafi 9122 . . 3 ((Fun (1st ā†¾ (V Ɨ V)) āˆ§ (š¹ supp š‘) āˆˆ Fin) ā†’ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) āˆˆ Fin)
1915, 17, 18syl2anc 585 . 2 (šœ‘ ā†’ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) āˆˆ Fin)
20 ovexd 7393 . . . 4 ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ (š‘¦š¹š¶) āˆˆ V)
2120, 4fmptd 7063 . . 3 (šœ‘ ā†’ šŗ:š“āŸ¶V)
22 eldif 3921 . . . 4 (š‘¦ āˆˆ (š“ āˆ– ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘))) ā†” (š‘¦ āˆˆ š“ āˆ§ Ā¬ š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘))))
23 simplr 768 . . . . . . . . . . 11 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ š‘¦ āˆˆ š“)
24 fsuppcurry2.c . . . . . . . . . . . 12 (šœ‘ ā†’ š¶ āˆˆ šµ)
2524ad2antrr 725 . . . . . . . . . . 11 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ š¶ āˆˆ šµ)
2623, 25opelxpd 5672 . . . . . . . . . 10 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ āŸØš‘¦, š¶āŸ© āˆˆ (š“ Ɨ šµ))
27 df-ov 7361 . . . . . . . . . . 11 (š‘¦š¹š¶) = (š¹ā€˜āŸØš‘¦, š¶āŸ©)
28 ovexd 7393 . . . . . . . . . . . . 13 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (š‘¦š¹š¶) āˆˆ V)
291, 2, 23, 28fvmptd3 6972 . . . . . . . . . . . 12 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (šŗā€˜š‘¦) = (š‘¦š¹š¶))
30 simpr 486 . . . . . . . . . . . . 13 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ Ā¬ (šŗā€˜š‘¦) = š‘)
3130neqned 2947 . . . . . . . . . . . 12 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (šŗā€˜š‘¦) ā‰  š‘)
3229, 31eqnetrrd 3009 . . . . . . . . . . 11 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (š‘¦š¹š¶) ā‰  š‘)
3327, 32eqnetrrid 3016 . . . . . . . . . 10 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (š¹ā€˜āŸØš‘¦, š¶āŸ©) ā‰  š‘)
34 fsuppcurry2.f . . . . . . . . . . . 12 (šœ‘ ā†’ š¹ Fn (š“ Ɨ šµ))
35 fsuppcurry2.b . . . . . . . . . . . . 13 (šœ‘ ā†’ šµ āˆˆ š‘Š)
365, 35xpexd 7686 . . . . . . . . . . . 12 (šœ‘ ā†’ (š“ Ɨ šµ) āˆˆ V)
37 elsuppfn 8103 . . . . . . . . . . . 12 ((š¹ Fn (š“ Ɨ šµ) āˆ§ (š“ Ɨ šµ) āˆˆ V āˆ§ š‘ āˆˆ š‘ˆ) ā†’ (āŸØš‘¦, š¶āŸ© āˆˆ (š¹ supp š‘) ā†” (āŸØš‘¦, š¶āŸ© āˆˆ (š“ Ɨ šµ) āˆ§ (š¹ā€˜āŸØš‘¦, š¶āŸ©) ā‰  š‘)))
3834, 36, 10, 37syl3anc 1372 . . . . . . . . . . 11 (šœ‘ ā†’ (āŸØš‘¦, š¶āŸ© āˆˆ (š¹ supp š‘) ā†” (āŸØš‘¦, š¶āŸ© āˆˆ (š“ Ɨ šµ) āˆ§ (š¹ā€˜āŸØš‘¦, š¶āŸ©) ā‰  š‘)))
3938ad2antrr 725 . . . . . . . . . 10 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (āŸØš‘¦, š¶āŸ© āˆˆ (š¹ supp š‘) ā†” (āŸØš‘¦, š¶āŸ© āˆˆ (š“ Ɨ šµ) āˆ§ (š¹ā€˜āŸØš‘¦, š¶āŸ©) ā‰  š‘)))
4026, 33, 39mpbir2and 712 . . . . . . . . 9 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ āŸØš‘¦, š¶āŸ© āˆˆ (š¹ supp š‘))
41 simpr 486 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ š‘§ = āŸØš‘¦, š¶āŸ©)
4241fveq2d 6847 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ ((1st ā†¾ (V Ɨ V))ā€˜š‘§) = ((1st ā†¾ (V Ɨ V))ā€˜āŸØš‘¦, š¶āŸ©))
43 xpss 5650 . . . . . . . . . . . 12 (š“ Ɨ šµ) āŠ† (V Ɨ V)
4426adantr 482 . . . . . . . . . . . 12 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ āŸØš‘¦, š¶āŸ© āˆˆ (š“ Ɨ šµ))
4543, 44sselid 3943 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ āŸØš‘¦, š¶āŸ© āˆˆ (V Ɨ V))
4645fvresd 6863 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ ((1st ā†¾ (V Ɨ V))ā€˜āŸØš‘¦, š¶āŸ©) = (1st ā€˜āŸØš‘¦, š¶āŸ©))
4723adantr 482 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ š‘¦ āˆˆ š“)
4825adantr 482 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ š¶ āˆˆ šµ)
49 op1stg 7934 . . . . . . . . . . 11 ((š‘¦ āˆˆ š“ āˆ§ š¶ āˆˆ šµ) ā†’ (1st ā€˜āŸØš‘¦, š¶āŸ©) = š‘¦)
5047, 48, 49syl2anc 585 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ (1st ā€˜āŸØš‘¦, š¶āŸ©) = š‘¦)
5142, 46, 503eqtrd 2777 . . . . . . . . 9 ((((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) āˆ§ š‘§ = āŸØš‘¦, š¶āŸ©) ā†’ ((1st ā†¾ (V Ɨ V))ā€˜š‘§) = š‘¦)
5240, 51rspcedeq1vd 3585 . . . . . . . 8 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ āˆƒš‘§ āˆˆ (š¹ supp š‘)((1st ā†¾ (V Ɨ V))ā€˜š‘§) = š‘¦)
53 fofn 6759 . . . . . . . . . . . . 13 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
54 fnresin 31586 . . . . . . . . . . . . 13 (1st Fn V ā†’ (1st ā†¾ (V Ɨ V)) Fn (V āˆ© (V Ɨ V)))
5511, 53, 54mp2b 10 . . . . . . . . . . . 12 (1st ā†¾ (V Ɨ V)) Fn (V āˆ© (V Ɨ V))
56 ssv 3969 . . . . . . . . . . . . . 14 (V Ɨ V) āŠ† V
57 sseqin2 4176 . . . . . . . . . . . . . 14 ((V Ɨ V) āŠ† V ā†” (V āˆ© (V Ɨ V)) = (V Ɨ V))
5856, 57mpbi 229 . . . . . . . . . . . . 13 (V āˆ© (V Ɨ V)) = (V Ɨ V)
5958fneq2i 6601 . . . . . . . . . . . 12 ((1st ā†¾ (V Ɨ V)) Fn (V āˆ© (V Ɨ V)) ā†” (1st ā†¾ (V Ɨ V)) Fn (V Ɨ V))
6055, 59mpbi 229 . . . . . . . . . . 11 (1st ā†¾ (V Ɨ V)) Fn (V Ɨ V)
6160a1i 11 . . . . . . . . . 10 (šœ‘ ā†’ (1st ā†¾ (V Ɨ V)) Fn (V Ɨ V))
62 suppssdm 8109 . . . . . . . . . . . 12 (š¹ supp š‘) āŠ† dom š¹
6334fndmd 6608 . . . . . . . . . . . 12 (šœ‘ ā†’ dom š¹ = (š“ Ɨ šµ))
6462, 63sseqtrid 3997 . . . . . . . . . . 11 (šœ‘ ā†’ (š¹ supp š‘) āŠ† (š“ Ɨ šµ))
6564, 43sstrdi 3957 . . . . . . . . . 10 (šœ‘ ā†’ (š¹ supp š‘) āŠ† (V Ɨ V))
6661, 65fvelimabd 6916 . . . . . . . . 9 (šœ‘ ā†’ (š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) ā†” āˆƒš‘§ āˆˆ (š¹ supp š‘)((1st ā†¾ (V Ɨ V))ā€˜š‘§) = š‘¦))
6766ad2antrr 725 . . . . . . . 8 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ (š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) ā†” āˆƒš‘§ āˆˆ (š¹ supp š‘)((1st ā†¾ (V Ɨ V))ā€˜š‘§) = š‘¦))
6852, 67mpbird 257 . . . . . . 7 (((šœ‘ āˆ§ š‘¦ āˆˆ š“) āˆ§ Ā¬ (šŗā€˜š‘¦) = š‘) ā†’ š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)))
6968ex 414 . . . . . 6 ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ (Ā¬ (šŗā€˜š‘¦) = š‘ ā†’ š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘))))
7069con1d 145 . . . . 5 ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ (Ā¬ š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) ā†’ (šŗā€˜š‘¦) = š‘))
7170impr 456 . . . 4 ((šœ‘ āˆ§ (š‘¦ āˆˆ š“ āˆ§ Ā¬ š‘¦ āˆˆ ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)))) ā†’ (šŗā€˜š‘¦) = š‘)
7222, 71sylan2b 595 . . 3 ((šœ‘ āˆ§ š‘¦ āˆˆ (š“ āˆ– ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)))) ā†’ (šŗā€˜š‘¦) = š‘)
7321, 72suppss 8126 . 2 (šœ‘ ā†’ (šŗ supp š‘) āŠ† ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)))
74 suppssfifsupp 9325 . 2 (((šŗ āˆˆ V āˆ§ Fun šŗ āˆ§ š‘ āˆˆ š‘ˆ) āˆ§ (((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)) āˆˆ Fin āˆ§ (šŗ supp š‘) āŠ† ((1st ā†¾ (V Ɨ V)) ā€œ (š¹ supp š‘)))) ā†’ šŗ finSupp š‘)
757, 9, 10, 19, 73, 74syl32anc 1379 1 (šœ‘ ā†’ šŗ finSupp š‘)
Colors of variables: wff setvar class
Syntax hints:  Ā¬ wn 3   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397   = wceq 1542   āˆˆ wcel 2107   ā‰  wne 2940  āˆƒwrex 3070  Vcvv 3444   āˆ– cdif 3908   āˆ© cin 3910   āŠ† wss 3911  āŸØcop 4593   class class class wbr 5106   ā†¦ cmpt 5189   Ɨ cxp 5632  dom cdm 5634   ā†¾ cres 5636   ā€œ cima 5637  Fun wfun 6491   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  (class class class)co 7358  1st c1st 7920   supp csupp 8093  Fincfn 8886   finSupp cfsupp 9308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-supp 8094  df-1o 8413  df-en 8887  df-fin 8890  df-fsupp 9309
This theorem is referenced by: (None)
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