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Theorem fpwrelmapffs 30396
Description: Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypotheses
Ref Expression
fpwrelmap.1 𝐴 ∈ V
fpwrelmap.2 𝐵 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
fpwrelmapffs.1 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}
Assertion
Ref Expression
fpwrelmapffs (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑓)   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmapffs
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . . 4 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
3 fpwrelmap.2 . . . . . 6 𝐵 ∈ V
42, 3, 1fpwrelmap 30395 . . . . 5 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
54a1i 11 . . . 4 (⊤ → 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
6 simpl 483 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 ∈ (𝒫 𝐵m 𝐴))
73pwex 5272 . . . . . . . 8 𝒫 𝐵 ∈ V
87, 2elmap 8424 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
96, 8sylib 219 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓:𝐴⟶𝒫 𝐵)
10 simpr 485 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
112, 3, 9, 10fpwrelmapffslem 30394 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
12113adant1 1122 . . . 4 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
131, 5, 12f1oresrab 6881 . . 3 (⊤ → (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
1413mptru 1535 . 2 (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
15 fpwrelmapffs.1 . . . . 5 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}
162, 7maprnin 30393 . . . . . 6 ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin}
17 nfcv 2974 . . . . . . 7 𝑓((𝒫 𝐵 ∩ Fin) ↑m 𝐴)
18 nfrab1 3382 . . . . . . 7 𝑓{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin}
1917, 18rabeqf 3479 . . . . . 6 (((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin} → {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin})
2016, 19ax-mp 5 . . . . 5 {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin}
21 rabrab 3377 . . . . 5 {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
2215, 20, 213eqtri 2845 . . . 4 𝑆 = {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
23 dfin5 3941 . . . 4 (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
24 f1oeq23 6600 . . . 4 ((𝑆 = {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)} ∧ (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}) → ((𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) ↔ (𝑀𝑆):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}))
2522, 23, 24mp2an 688 . . 3 ((𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) ↔ (𝑀𝑆):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
2622reseq2i 5843 . . . 4 (𝑀𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
27 f1oeq1 6597 . . . 4 ((𝑀𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}) → ((𝑀𝑆):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}))
2826, 27ax-mp 5 . . 3 ((𝑀𝑆):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
2925, 28bitr2i 277 . 2 ((𝑀 ↾ {𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin))
3014, 29mpbi 231 1 (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wtru 1529  wcel 2105  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {copab 5119  cmpt 5137   × cxp 5546  ran crn 5549  cres 5550  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145   supp csupp 7819  m cmap 8395  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-fin 8501  df-card 9356  df-acn 9359  df-ac 9530
This theorem is referenced by:  eulerpartlem1  31524
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