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Theorem fsuppcurry2 32743
Description: Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
Hypotheses
Ref Expression
fsuppcurry2.g 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
fsuppcurry2.z (𝜑𝑍𝑈)
fsuppcurry2.a (𝜑𝐴𝑉)
fsuppcurry2.b (𝜑𝐵𝑊)
fsuppcurry2.f (𝜑𝐹 Fn (𝐴 × 𝐵))
fsuppcurry2.c (𝜑𝐶𝐵)
fsuppcurry2.1 (𝜑𝐹 finSupp 𝑍)
Assertion
Ref Expression
fsuppcurry2 (𝜑𝐺 finSupp 𝑍)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑈(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem fsuppcurry2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsuppcurry2.g . . . 4 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶))
2 oveq1 7437 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
32cbvmptv 5260 . . . 4 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
41, 3eqtri 2762 . . 3 𝐺 = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
5 fsuppcurry2.a . . . 4 (𝜑𝐴𝑉)
65mptexd 7243 . . 3 (𝜑 → (𝑦𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
74, 6eqeltrid 2842 . 2 (𝜑𝐺 ∈ V)
81funmpt2 6606 . . 3 Fun 𝐺
98a1i 11 . 2 (𝜑 → Fun 𝐺)
10 fsuppcurry2.z . 2 (𝜑𝑍𝑈)
11 fo1st 8032 . . . . 5 1st :V–onto→V
12 fofun 6821 . . . . 5 (1st :V–onto→V → Fun 1st )
1311, 12ax-mp 5 . . . 4 Fun 1st
14 funres 6609 . . . 4 (Fun 1st → Fun (1st ↾ (V × V)))
1513, 14mp1i 13 . . 3 (𝜑 → Fun (1st ↾ (V × V)))
16 fsuppcurry2.1 . . . 4 (𝜑𝐹 finSupp 𝑍)
1716fsuppimpd 9406 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18 imafi 9350 . . 3 ((Fun (1st ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
1915, 17, 18syl2anc 584 . 2 (𝜑 → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20 ovexd 7465 . . . 4 ((𝜑𝑦𝐴) → (𝑦𝐹𝐶) ∈ V)
2120, 4fmptd 7133 . . 3 (𝜑𝐺:𝐴⟶V)
22 eldif 3972 . . . 4 (𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
23 simplr 769 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝑦𝐴)
24 fsuppcurry2.c . . . . . . . . . . . 12 (𝜑𝐶𝐵)
2524ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝐶𝐵)
2623, 25opelxpd 5727 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27 df-ov 7433 . . . . . . . . . . 11 (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28 ovexd 7465 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
291, 2, 23, 28fvmptd3 7038 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = (𝑦𝐹𝐶))
30 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ¬ (𝐺𝑦) = 𝑍)
3130neqned 2944 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) ≠ 𝑍)
3229, 31eqnetrrd 3006 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
3327, 32eqnetrrid 3013 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)
34 fsuppcurry2.f . . . . . . . . . . . 12 (𝜑𝐹 Fn (𝐴 × 𝐵))
35 fsuppcurry2.b . . . . . . . . . . . . 13 (𝜑𝐵𝑊)
365, 35xpexd 7769 . . . . . . . . . . . 12 (𝜑 → (𝐴 × 𝐵) ∈ V)
37 elsuppfn 8193 . . . . . . . . . . . 12 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ∈ V ∧ 𝑍𝑈) → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
3834, 36, 10, 37syl3anc 1370 . . . . . . . . . . 11 (𝜑 → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
3938ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
4026, 33, 39mpbir2and 713 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍))
41 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
4241fveq2d 6910 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43 xpss 5704 . . . . . . . . . . . 12 (𝐴 × 𝐵) ⊆ (V × V)
4426adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
4543, 44sselid 3992 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
4645fvresd 6926 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
4723adantr 480 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦𝐴)
4825adantr 480 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶𝐵)
49 op1stg 8024 . . . . . . . . . . 11 ((𝑦𝐴𝐶𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5047, 48, 49syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5142, 46, 503eqtrd 2778 . . . . . . . . 9 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
5240, 51rspcedeq1vd 3628 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53 fofn 6822 . . . . . . . . . . . . 13 (1st :V–onto→V → 1st Fn V)
54 fnresin 32642 . . . . . . . . . . . . 13 (1st Fn V → (1st ↾ (V × V)) Fn (V ∩ (V × V)))
5511, 53, 54mp2b 10 . . . . . . . . . . . 12 (1st ↾ (V × V)) Fn (V ∩ (V × V))
56 ssv 4019 . . . . . . . . . . . . . 14 (V × V) ⊆ V
57 sseqin2 4230 . . . . . . . . . . . . . 14 ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
5856, 57mpbi 230 . . . . . . . . . . . . 13 (V ∩ (V × V)) = (V × V)
5958fneq2i 6666 . . . . . . . . . . . 12 ((1st ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (1st ↾ (V × V)) Fn (V × V))
6055, 59mpbi 230 . . . . . . . . . . 11 (1st ↾ (V × V)) Fn (V × V)
6160a1i 11 . . . . . . . . . 10 (𝜑 → (1st ↾ (V × V)) Fn (V × V))
62 suppssdm 8200 . . . . . . . . . . . 12 (𝐹 supp 𝑍) ⊆ dom 𝐹
6334fndmd 6673 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
6462, 63sseqtrid 4047 . . . . . . . . . . 11 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
6564, 43sstrdi 4007 . . . . . . . . . 10 (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
6661, 65fvelimabd 6981 . . . . . . . . 9 (𝜑 → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
6766ad2antrr 726 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
6852, 67mpbird 257 . . . . . . 7 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
6968ex 412 . . . . . 6 ((𝜑𝑦𝐴) → (¬ (𝐺𝑦) = 𝑍𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
7069con1d 145 . . . . 5 ((𝜑𝑦𝐴) → (¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺𝑦) = 𝑍))
7170impr 454 . . . 4 ((𝜑 ∧ (𝑦𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺𝑦) = 𝑍)
7222, 71sylan2b 594 . . 3 ((𝜑𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺𝑦) = 𝑍)
7321, 72suppss 8217 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74 suppssfifsupp 9417 . 2 (((𝐺 ∈ V ∧ Fun 𝐺𝑍𝑈) ∧ (((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → 𝐺 finSupp 𝑍)
757, 9, 10, 19, 73, 74syl32anc 1377 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cdif 3959  cin 3961  wss 3962  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  ontowfo 6560  cfv 6562  (class class class)co 7430  1st c1st 8010   supp csupp 8183  Fincfn 8983   finSupp cfsupp 9398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-supp 8184  df-1o 8504  df-en 8984  df-dom 8985  df-fin 8987  df-fsupp 9399
This theorem is referenced by: (None)
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