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Theorem fpwrelmapffs 32226
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 𝐡 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))})
fpwrelmapffs.1 𝑆 = {𝑓 ∈ ((𝒫 𝐡 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp βˆ…) ∈ Fin}
Assertion
Ref Expression
fpwrelmapffs (𝑀 β†Ύ 𝑆):𝑆–1-1-ontoβ†’(𝒫 (𝐴 Γ— 𝐡) ∩ Fin)
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐴   𝐡,𝑓,π‘₯,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦,𝑓)   𝑀(π‘₯,𝑦,𝑓)

Proof of Theorem fpwrelmapffs
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . . 4 𝑀 = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
3 fpwrelmap.2 . . . . . 6 𝐡 ∈ V
42, 3, 1fpwrelmap 32225 . . . . 5 𝑀:(𝒫 𝐡 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 Γ— 𝐡)
54a1i 11 . . . 4 (⊀ β†’ 𝑀:(𝒫 𝐡 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 Γ— 𝐡))
6 simpl 481 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))}) β†’ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴))
73pwex 5377 . . . . . . . 8 𝒫 𝐡 ∈ V
87, 2elmap 8867 . . . . . . 7 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↔ 𝑓:π΄βŸΆπ’« 𝐡)
96, 8sylib 217 . . . . . 6 ((𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))}) β†’ 𝑓:π΄βŸΆπ’« 𝐡)
10 simpr 483 . . . . . 6 ((𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))}) β†’ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))})
112, 3, 9, 10fpwrelmapffslem 32224 . . . . 5 ((𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))}) β†’ (π‘Ÿ ∈ Fin ↔ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)))
12113adant1 1128 . . . 4 ((⊀ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∧ π‘Ÿ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ (π‘“β€˜π‘₯))}) β†’ (π‘Ÿ ∈ Fin ↔ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)))
131, 5, 12f1oresrab 7126 . . 3 (⊀ β†’ (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin})
1413mptru 1546 . 2 (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin}
15 fpwrelmapffs.1 . . . . 5 𝑆 = {𝑓 ∈ ((𝒫 𝐡 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp βˆ…) ∈ Fin}
162, 7maprnin 32223 . . . . . 6 ((𝒫 𝐡 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin}
17 nfcv 2901 . . . . . . 7 Ⅎ𝑓((𝒫 𝐡 ∩ Fin) ↑m 𝐴)
18 nfrab1 3449 . . . . . . 7 Ⅎ𝑓{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin}
1917, 18rabeqf 3464 . . . . . 6 (((𝒫 𝐡 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin} β†’ {𝑓 ∈ ((𝒫 𝐡 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp βˆ…) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin} ∣ (𝑓 supp βˆ…) ∈ Fin})
2016, 19ax-mp 5 . . . . 5 {𝑓 ∈ ((𝒫 𝐡 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp βˆ…) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin} ∣ (𝑓 supp βˆ…) ∈ Fin}
21 rabrab 3453 . . . . 5 {𝑓 ∈ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ ran 𝑓 βŠ† Fin} ∣ (𝑓 supp βˆ…) ∈ Fin} = {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}
2215, 20, 213eqtri 2762 . . . 4 𝑆 = {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}
23 dfin5 3955 . . . 4 (𝒫 (𝐴 Γ— 𝐡) ∩ Fin) = {π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin}
24 f1oeq23 6823 . . . 4 ((𝑆 = {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)} ∧ (𝒫 (𝐴 Γ— 𝐡) ∩ Fin) = {π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin}) β†’ ((𝑀 β†Ύ 𝑆):𝑆–1-1-ontoβ†’(𝒫 (𝐴 Γ— 𝐡) ∩ Fin) ↔ (𝑀 β†Ύ 𝑆):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin}))
2522, 23, 24mp2an 688 . . 3 ((𝑀 β†Ύ 𝑆):𝑆–1-1-ontoβ†’(𝒫 (𝐴 Γ— 𝐡) ∩ Fin) ↔ (𝑀 β†Ύ 𝑆):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin})
2622reseq2i 5977 . . . 4 (𝑀 β†Ύ 𝑆) = (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)})
27 f1oeq1 6820 . . . 4 ((𝑀 β†Ύ 𝑆) = (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}) β†’ ((𝑀 β†Ύ 𝑆):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin} ↔ (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin}))
2826, 27ax-mp 5 . . 3 ((𝑀 β†Ύ 𝑆):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin} ↔ (𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin})
2925, 28bitr2i 275 . 2 ((𝑀 β†Ύ {𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ∣ (ran 𝑓 βŠ† Fin ∧ (𝑓 supp βˆ…) ∈ Fin)}–1-1-ontoβ†’{π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐡) ∣ π‘Ÿ ∈ Fin} ↔ (𝑀 β†Ύ 𝑆):𝑆–1-1-ontoβ†’(𝒫 (𝐴 Γ— 𝐡) ∩ Fin))
3014, 29mpbi 229 1 (𝑀 β†Ύ 𝑆):𝑆–1-1-ontoβ†’(𝒫 (𝐴 Γ— 𝐡) ∩ Fin)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  βŠ€wtru 1540   ∈ wcel 2104  {crab 3430  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {copab 5209   ↦ cmpt 5230   Γ— cxp 5673  ran crn 5676   β†Ύ cres 5677  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   supp csupp 8148   ↑m cmap 8822  Fincfn 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-card 9936  df-acn 9939  df-ac 10113
This theorem is referenced by:  eulerpartlem1  33664
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