Theorem fpwrelmapffs 33368

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 33367	. . . . 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	3	pwex 5341	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
7	6, 2	elmap 8899	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8	7	birani 509	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓:𝐴⟶𝒫 𝐵)
9		simpr 490	. . . . . 6 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})
10	2, 3, 8, 9	fpwrelmapffslem 33366	. . . . 5 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
11	10	3adant1 1148	. . . 4 ⊢ ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ Fin ↔ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)))
12	1, 5, 11	f1oresrab 7128	. . 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})
13	12	mptru 1576	. 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}
14		fpwrelmapffs.1	. . . . 5 ⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}
15	2, 6	maprnin 33365	. . . . . 6 ⊢ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
16		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑓((𝒫 𝐵 ∩ Fin) ↑m 𝐴)
17		nfrab1 3431	. . . . . . 7 ⊢ Ⅎ𝑓{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
18	16, 17	rabeqf 3445	. . . . . 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})
19	15, 18	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}
20		rabrab 3435	. . . . 5 ⊢ {𝑓 ∈ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin} ∣ (𝑓 supp ∅) ∈ Fin} = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
21	14, 19, 20	3eqtri 2787	. . . 4 ⊢ 𝑆 = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
22		dfin5 4092	. . . 4 ⊢ (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
23		f1oeq23 6815	. . . 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}))
24	21, 22, 23	mp2an 705	. . 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})
25	21	reseq2i 5966	. . . 4 ⊢ (𝑀 ↾ 𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
26		f1oeq1 6812	. . . 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}))
27	25, 26	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})
28	24, 27	bitr2i 279	. 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))
29	13, 28	mpbi 233	1 ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {copab 5166   ↦ cmpt 5185   × cxp 5648  ran crn 5651   ↾ cres 5652  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   supp csupp 8177   ↑_m cmap 8847  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-fin 8977  df-card 10020  df-acn 10023

This theorem is used by:  eulerpartlem1  35041