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Theorem fpwrelmapffs 29849
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 29848 . . . . 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 468 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
73pwex 4981 . . . . . . . 8 𝒫 𝐵 ∈ V
87, 2elmap 8038 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
96, 8sylib 208 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓:𝐴⟶𝒫 𝐵)
10 simpr 471 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
112, 3, 9, 10fpwrelmapffslem 29847 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
12113adant1 1124 . . . 4 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
131, 5, 12f1oresrab 6537 . . 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})
1413trud 1641 . 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 29846 . . . . . 6 ((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴) = {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ ran 𝑓 ⊆ Fin}
17 nfcv 2913 . . . . . . 7 𝑓((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴)
18 nfrab1 3271 . . . . . . 7 𝑓{𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ ran 𝑓 ⊆ Fin}
1917, 18rabeqf 3340 . . . . . 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 3265 . . . . 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 2797 . . . 4 𝑆 = {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
23 dfin5 3731 . . . 4 (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
24 f1oeq23 6271 . . . 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 672 . . 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 5531 . . . 4 (𝑀𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
27 f1oeq1 6268 . . . 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 265 . 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 220 1 (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wtru 1632  wcel 2145  {crab 3065  Vcvv 3351  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {copab 4846  cmpt 4863   × cxp 5247  ran crn 5250  cres 5251  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793   supp csupp 7446  𝑚 cmap 8009  Fincfn 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-ac2 9487
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-fin 8113  df-card 8965  df-acn 8968  df-ac 9139
This theorem is referenced by:  eulerpartlem1  30769
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