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Theorem fsuppcurry2 31643
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 7364 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐹𝐶) = (𝑦𝐹𝐶))
32cbvmptv 5218 . . . 4 (𝑥𝐴 ↦ (𝑥𝐹𝐶)) = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
41, 3eqtri 2764 . . 3 𝐺 = (𝑦𝐴 ↦ (𝑦𝐹𝐶))
5 fsuppcurry2.a . . . 4 (𝜑𝐴𝑉)
65mptexd 7174 . . 3 (𝜑 → (𝑦𝐴 ↦ (𝑦𝐹𝐶)) ∈ V)
74, 6eqeltrid 2842 . 2 (𝜑𝐺 ∈ V)
81funmpt2 6540 . . 3 Fun 𝐺
98a1i 11 . 2 (𝜑 → Fun 𝐺)
10 fsuppcurry2.z . 2 (𝜑𝑍𝑈)
11 fo1st 7941 . . . . 5 1st :V–onto→V
12 fofun 6757 . . . . 5 (1st :V–onto→V → Fun 1st )
1311, 12ax-mp 5 . . . 4 Fun 1st
14 funres 6543 . . . 4 (Fun 1st → Fun (1st ↾ (V × V)))
1513, 14mp1i 13 . . 3 (𝜑 → Fun (1st ↾ (V × V)))
16 fsuppcurry2.1 . . . 4 (𝜑𝐹 finSupp 𝑍)
1716fsuppimpd 9312 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
18 imafi 9119 . . 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 7392 . . . 4 ((𝜑𝑦𝐴) → (𝑦𝐹𝐶) ∈ V)
2120, 4fmptd 7062 . . 3 (𝜑𝐺:𝐴⟶V)
22 eldif 3920 . . . 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 5671 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
27 df-ov 7360 . . . . . . . . . . 11 (𝑦𝐹𝐶) = (𝐹‘⟨𝑦, 𝐶⟩)
28 ovexd 7392 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ∈ V)
291, 2, 23, 28fvmptd3 6971 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = (𝑦𝐹𝐶))
30 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ¬ (𝐺𝑦) = 𝑍)
3130neqned 2950 . . . . . . . . . . . 12 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐺𝑦) ≠ 𝑍)
3229, 31eqnetrrd 3012 . . . . . . . . . . 11 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝑦𝐹𝐶) ≠ 𝑍)
3327, 32eqnetrrid 3019 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → (𝐹‘⟨𝑦, 𝐶⟩) ≠ 𝑍)
34 fsuppcurry2.f . . . . . . . . . . . 12 (𝜑𝐹 Fn (𝐴 × 𝐵))
35 fsuppcurry2.b . . . . . . . . . . . . 13 (𝜑𝐵𝑊)
365, 35xpexd 7685 . . . . . . . . . . . 12 (𝜑 → (𝐴 × 𝐵) ∈ V)
37 elsuppfn 8102 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑧 = ⟨𝑦, 𝐶⟩)
4241fveq2d 6846 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩))
43 xpss 5649 . . . . . . . . . . . 12 (𝐴 × 𝐵) ⊆ (V × V)
4426adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (𝐴 × 𝐵))
4543, 44sselid 3942 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ⟨𝑦, 𝐶⟩ ∈ (V × V))
4645fvresd 6862 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘⟨𝑦, 𝐶⟩) = (1st ‘⟨𝑦, 𝐶⟩))
4723adantr 481 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝑦𝐴)
4825adantr 481 . . . . . . . . . . 11 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → 𝐶𝐵)
49 op1stg 7933 . . . . . . . . . . 11 ((𝑦𝐴𝐶𝐵) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5047, 48, 49syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → (1st ‘⟨𝑦, 𝐶⟩) = 𝑦)
5142, 46, 503eqtrd 2780 . . . . . . . . 9 ((((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) ∧ 𝑧 = ⟨𝑦, 𝐶⟩) → ((1st ↾ (V × V))‘𝑧) = 𝑦)
5240, 51rspcedeq1vd 3586 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ ¬ (𝐺𝑦) = 𝑍) → ∃𝑧 ∈ (𝐹 supp 𝑍)((1st ↾ (V × V))‘𝑧) = 𝑦)
53 fofn 6758 . . . . . . . . . . . . 13 (1st :V–onto→V → 1st Fn V)
54 fnresin 31540 . . . . . . . . . . . . 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 3968 . . . . . . . . . . . . . 14 (V × V) ⊆ V
57 sseqin2 4175 . . . . . . . . . . . . . 14 ((V × V) ⊆ V ↔ (V ∩ (V × V)) = (V × V))
5856, 57mpbi 229 . . . . . . . . . . . . 13 (V ∩ (V × V)) = (V × V)
5958fneq2i 6600 . . . . . . . . . . . 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 8108 . . . . . . . . . . . 12 (𝐹 supp 𝑍) ⊆ dom 𝐹
6334fndmd 6607 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = (𝐴 × 𝐵))
6462, 63sseqtrid 3996 . . . . . . . . . . 11 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐴 × 𝐵))
6564, 43sstrdi 3956 . . . . . . . . . 10 (𝜑 → (𝐹 supp 𝑍) ⊆ (V × V))
6661, 65fvelimabd 6915 . . . . . . . . 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 413 . . . . . 6 ((𝜑𝑦𝐴) → (¬ (𝐺𝑦) = 𝑍𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍))))
7069con1d 145 . . . . 5 ((𝜑𝑦𝐴) → (¬ 𝑦 ∈ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)) → (𝐺𝑦) = 𝑍))
7170impr 455 . . . 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 8125 . 2 (𝜑 → (𝐺 supp 𝑍) ⊆ ((1st ↾ (V × V)) “ (𝐹 supp 𝑍)))
74 suppssfifsupp 9320 . 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 1378 1 (𝜑𝐺 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  wss 3910  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  ontowfo 6494  cfv 6496  (class class class)co 7357  1st c1st 7919   supp csupp 8092  Fincfn 8883   finSupp cfsupp 9305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-supp 8093  df-1o 8412  df-en 8884  df-fin 8887  df-fsupp 9306
This theorem is referenced by: (None)
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