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Theorem fsuppcurry2 32590
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 7426 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
32cbvmptv 5262 . . . 4 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
41, 3eqtri 2753 . . 3 𝐺 = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
5 fsuppcurry2.a . . . 4 (𝜑𝐴𝑉)
65mptexd 7236 . . 3 (𝜑 → (𝑦𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
74, 6eqeltrid 2829 . 2 (𝜑𝐺 ∈ V)
81funmpt2 6593 . . 3 Fun 𝐺
98a1i 11 . 2 (𝜑 → Fun 𝐺)
10 fsuppcurry2.z . 2 (𝜑𝑍𝑈)
11 fo1st 8014 . . . . 5 1st :V–onto→V
12 fofun 6811 . . . . 5 (1st :V–onto→V → Fun 1st )
1311, 12ax-mp 5 . . . 4 Fun 1st
14 funres 6596 . . . 4 (Fun 1st → Fun (1st ↾ (V × V)))
1513, 14mp1i 13 . . 3 (𝜑 → Fun (1st ↾ (V × V)))
16 fsuppcurry2.1 . . . 4 (𝜑𝐹 finSupp 𝑍)
1716fsuppimpd 9395 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18 imafi 9200 . . 3 ((Fun (1st ↾ (V × V)) ∧ (𝐹 supp 𝑍) ∈ Fin) → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
1915, 17, 18syl2anc 582 . 2 (𝜑 → ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ∈ Fin)
20 ovexd 7454 . . . 4 ((𝜑𝑦𝐴) → (𝑦𝐹𝐶) ∈ V)
2120, 4fmptd 7123 . . 3 (𝜑𝐺:𝐴⟶V)
22 eldif 3954 . . . 4 (𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
23 simplr 767 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝑦𝐴)
24 fsuppcurry2.c . . . . . . . . . . . 12 (𝜑𝐶𝐵)
2524ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝐶𝐵)
2623, 25opelxpd 5717 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27 df-ov 7422 . . . . . . . . . . 11 (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28 ovexd 7454 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
291, 2, 23, 28fvmptd3 7027 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = (𝑦𝐹𝐶))
30 simpr 483 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ¬ (𝐺𝑦) = 𝑍)
3130neqned 2936 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) ≠ 𝑍)
3229, 31eqnetrrd 2998 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
3327, 32eqnetrrid 3005 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)
34 fsuppcurry2.f . . . . . . . . . . . 12 (𝜑𝐹 Fn (𝐴 × 𝐵))
35 fsuppcurry2.b . . . . . . . . . . . . 13 (𝜑𝐵𝑊)
365, 35xpexd 7754 . . . . . . . . . . . 12 (𝜑 → (𝐴 × 𝐵) ∈ V)
37 elsuppfn 8175 . . . . . . . . . . . 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 1368 . . . . . . . . . . 11 (𝜑 → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
3938ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍) ↔ (⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵) ∧ (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)))
4026, 33, 39mpbir2and 711 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐹 supp 𝑍))
41 simpr 483 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
4241fveq2d 6900 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43 xpss 5694 . . . . . . . . . . . 12 (𝐴 × 𝐵) ⊆ (V × V)
4426adantr 479 . . . . . . . . . . . 12 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
4543, 44sselid 3974 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
4645fvresd 6916 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
4723adantr 479 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦𝐴)
4825adantr 479 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶𝐵)
49 op1stg 8006 . . . . . . . . . . 11 ((𝑦𝐴𝐶𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5047, 48, 49syl2anc 582 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5142, 46, 503eqtrd 2769 . . . . . . . . 9 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
5240, 51rspcedeq1vd 3613 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53 fofn 6812 . . . . . . . . . . . . 13 (1st :V–onto→V → 1st Fn V)
54 fnresin 32492 . . . . . . . . . . . . 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 4001 . . . . . . . . . . . . . 14 (V × V) ⊆ V
57 sseqin2 4213 . . . . . . . . . . . . . 14 ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
5856, 57mpbi 229 . . . . . . . . . . . . 13 (V ∩ (V × V)) = (V × V)
5958fneq2i 6653 . . . . . . . . . . . 12 ((1st ↾ (V × V)) Fn (V ∩ (V × V)) ↔ (1st ↾ (V × V)) Fn (V × V))
6055, 59mpbi 229 . . . . . . . . . . 11 (1st ↾ (V × V)) Fn (V × V)
6160a1i 11 . . . . . . . . . 10 (𝜑 → (1st ↾ (V × V)) Fn (V × V))
62 suppssdm 8182 . . . . . . . . . . . 12 (𝐹 supp 𝑍) ⊆ dom 𝐹
6334fndmd 6660 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
6462, 63sseqtrid 4029 . . . . . . . . . . 11 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
6564, 43sstrdi 3989 . . . . . . . . . 10 (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
6661, 65fvelimabd 6971 . . . . . . . . 9 (𝜑 → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
6766ad2antrr 724 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) ↔ ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦))
6852, 67mpbird 256 . . . . . . 7 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
6968ex 411 . . . . . 6 ((𝜑𝑦𝐴) → (¬ (𝐺𝑦) = 𝑍𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
7069con1d 145 . . . . 5 ((𝜑𝑦𝐴) → (¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺𝑦) = 𝑍))
7170impr 453 . . . 4 ((𝜑 ∧ (𝑦𝐴 ∧ ¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺𝑦) = 𝑍)
7222, 71sylan2b 592 . . 3 ((𝜑𝑦 ∈ (𝐴 ∖ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))) → (𝐺𝑦) = 𝑍)
7321, 72suppss 8199 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74 suppssfifsupp 9405 . 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 1375 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wrex 3059  Vcvv 3461  cdif 3941  cin 3943  wss 3944  cop 4636   class class class wbr 5149  cmpt 5232   × cxp 5676  dom cdm 5678  cres 5680  cima 5681  Fun wfun 6543   Fn wfn 6544  ontowfo 6547  cfv 6549  (class class class)co 7419  1st c1st 7992   supp csupp 8165  Fincfn 8964   finSupp cfsupp 9387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-supp 8166  df-1o 8487  df-en 8965  df-fin 8968  df-fsupp 9388
This theorem is referenced by: (None)
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