Theorem fpwrelmapffs 32657

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 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
fpwrelmapffs.1	⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}

Assertion

Ref	Expression
fpwrelmapffs	⊢ (𝑀 ↾ 𝑆):𝑆–1-1-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.

Step	Hyp	Ref	Expression
1		fpwrelmap.3	. . . 4 ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
2		fpwrelmap.1	. . . . . 6 ⊢ 𝐴 ∈ V
3		fpwrelmap.2	. . . . . 6 ⊢ 𝐵 ∈ V
4	2, 3, 1	fpwrelmap 32656	. . . . 5 ⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
5	4	a1i 11	. . . 4 ⊢ (⊤ → 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
6		simpl 482	. . . . . . 7 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))
7	3	pwex 5335	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
8	7, 2	elmap 8844	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
9	6, 8	sylib 218	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓:𝐴⟶𝒫 𝐵)
10		simpr 484	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
11	2, 3, 9, 10	fpwrelmapffslem 32655	. . . . 5 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
12	11	3adant1 1130	. . . 4 ⊢ ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
13	1, 5, 12	f1oresrab 7099	. . 3 ⊢ (⊤ → (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
14	13	mptru 1547	. 2 ⊢ (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵 ↑m 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}
16	2, 7	maprnin 32654	. . . . . 6 ⊢ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
17		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑓((𝒫 𝐵 ∩ Fin) ↑m 𝐴)
18		nfrab1 3426	. . . . . . 7 ⊢ Ⅎ𝑓{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
19	17, 18	rabeqf 3440	. . . . . 6 ⊢ (((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin} → {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin})
20	16, 19	ax-mp 5	. . . . 5 ⊢ {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin}
21		rabrab 3430	. . . . 5 ⊢ {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
22	15, 20, 21	3eqtri 2756	. . . 4 ⊢ 𝑆 = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
23		dfin5 3922	. . . 4 ⊢ (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
24		f1oeq23 6791	. . . 4 ⊢ ((𝑆 = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)} ∧ (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}) → ((𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) ↔ (𝑀 ↾ 𝑆):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}))
25	22, 23, 24	mp2an 692	. . 3 ⊢ ((𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin) ↔ (𝑀 ↾ 𝑆):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
26	22	reseq2i 5947	. . . 4 ⊢ (𝑀 ↾ 𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
27		f1oeq1 6788	. . . 4 ⊢ ((𝑀 ↾ 𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}) → ((𝑀 ↾ 𝑆):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}))
28	26, 27	ax-mp 5	. . 3 ⊢ ((𝑀 ↾ 𝑆):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin})
29	25, 28	bitr2i 276	. 2 ⊢ ((𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}):{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}–1-1-onto→{𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin} ↔ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin))
30	14, 29	mpbi 230	1 ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {crab 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {copab 5169   ↦ cmpt 5188   × cxp 5636  ran crn 5639   ↾ cres 5640  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   supp csupp 8139   ↑_m cmap 8799  Fincfn 8918

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-ac2 10416

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-fin 8922  df-card 9892  df-acn 9895  df-ac 10069

This theorem is referenced by:  eulerpartlem1  34358