Theorem fpwrelmapffs 32794

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 32793	. . . . 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 5326	. . . . . . . 8 ⊢ 𝒫 𝐵 ∈ V
8	7, 2	elmap 8813	. . . . . . 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 32792	. . . . 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 1131	. . . 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 7074	. . 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 1549	. 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 32791	. . . . . 6 ⊢ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
17		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑓((𝒫 𝐵 ∩ Fin) ↑m 𝐴)
18		nfrab1 3420	. . . . . . 7 ⊢ Ⅎ𝑓{𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ Fin}
19	17, 18	rabeqf 3434	. . . . . 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 3424	. . . . 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 2764	. . . 4 ⊢ 𝑆 = {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)}
23		dfin5 3910	. . . 4 ⊢ (𝒫 (𝐴 × 𝐵) ∩ Fin) = {𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∣ 𝑟 ∈ Fin}
24		f1oeq23 6766	. . . 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 693	. . 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 5936	. . . 4 ⊢ (𝑀 ↾ 𝑆) = (𝑀 ↾ {𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ∣ (ran 𝑓 ⊆ Fin ∧ (𝑓 supp ∅) ∈ Fin)})
27		f1oeq1 6763	. . . 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 1542  ⊤wtru 1543   ∈ wcel 2114  {crab 3400  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  𝒫 cpw 4555  {copab 5161   ↦ cmpt 5180   × cxp 5623  ran crn 5626   ↾ cres 5627  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360   supp csupp 8104   ↑_m cmap 8767  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-ac2 10377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9855  df-acn 9858  df-ac 10030

This theorem is referenced by:  eulerpartlem1  34505