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Theorem fpwrelmapffs 29835
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 29834 . . . . 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 470 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
73pwex 5050 . . . . . . . 8 𝒫 𝐵 ∈ V
87, 2elmap 8117 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
96, 8sylib 209 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓:𝐴⟶𝒫 𝐵)
10 simpr 473 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
112, 3, 9, 10fpwrelmapffslem 29833 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
12113adant1 1153 . . . 4 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
131, 5, 12f1oresrab 6613 . . 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 1645 . 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 29832 . . . . . 6 ((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴) = {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ ran 𝑓 ⊆ Fin}
17 nfcv 2948 . . . . . . 7 𝑓((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴)
18 nfrab1 3311 . . . . . . 7 𝑓{𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ ran 𝑓 ⊆ Fin}
1917, 18rabeqf 3380 . . . . . 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 3305 . . . . 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 2832 . . . 4 𝑆 = {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
23 dfin5 3777 . . . 4 (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
24 f1oeq23 6342 . . . 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 675 . . 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 5594 . . . 4 (𝑀𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
27 f1oeq1 6339 . . . 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 267 . 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 221 1 (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wtru 1638  wcel 2156  {crab 3100  Vcvv 3391  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  {copab 4906  cmpt 4923   × cxp 5309  ran crn 5312  cres 5313  wf 6093  1-1-ontowf1o 6096  cfv 6097  (class class class)co 6870   supp csupp 7525  𝑚 cmap 8088  Fincfn 8188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-ac2 9566
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-supp 7526  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-fin 8192  df-card 9044  df-acn 9047  df-ac 9218
This theorem is referenced by:  eulerpartlem1  30753
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