Theorem prproropf1o 47617

Description: There is a bijection between the set of proper pairs and the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component. (Contributed by AV, 15-Mar-2023.)

Hypotheses

Ref	Expression
prproropf1o.o	⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
prproropf1o.p	⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
prproropf1o.f	⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)

Assertion

Ref	Expression
prproropf1o	⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂)

Distinct variable groups:   𝑉,𝑝   𝑂,𝑝   𝑃,𝑝   𝑅,𝑝

Allowed substitution hint:   𝐹(𝑝)

Proof of Theorem prproropf1o

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prproropf1o.o	. . . . 5 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
2		prproropf1o.p	. . . . 5 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2}
3	1, 2	prproropf1olem2 47614	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃) → ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩ ∈ 𝑂)
4		prproropf1o.f	. . . . 5 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩)
5		infeq1 9361	. . . . . . 7 ⊢ (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅))
6		supeq1 9329	. . . . . . 7 ⊢ (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅))
7	5, 6	opeq12d 4830	. . . . . 6 ⊢ (𝑝 = 𝑤 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
8	7	cbvmptv 5193	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↦ ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩) = (𝑤 ∈ 𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
9	4, 8	eqtri 2754	. . . 4 ⊢ 𝐹 = (𝑤 ∈ 𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
10	3, 9	fmptd 7047	. . 3 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃⟶𝑂)
11		3ancomb 1098	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))
12		3anass 1094	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)))
13	11, 12	bitri 275	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)))
14	1, 2, 4	prproropf1olem4 47616	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
15	13, 14	sylbir 235	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
16	15	ralrimivva 3175	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
17		dff13 7188	. . 3 ⊢ (𝐹:𝑃–1-1→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
18	10, 16, 17	sylanbrc 583	. 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1→𝑂)
19	1, 2	prproropf1olem1 47613	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → {(1st ‘𝑤), (2nd ‘𝑤)} ∈ 𝑃)
20		fveq2 6822	. . . . . . 7 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝐹‘𝑧) = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
21	20	eqeq2d 2742	. . . . . 6 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})))
22	21	adantl 481	. . . . 5 ⊢ (((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) ∧ 𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)}) → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})))
23	1, 2, 4	prproropf1olem3 47615	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}) = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
24	1	prproropf1olem0 47612	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑂 ↔ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∧ ((1st ‘𝑤) ∈ 𝑉 ∧ (2nd ‘𝑤) ∈ 𝑉) ∧ (1st ‘𝑤)𝑅(2nd ‘𝑤)))
25	24	simp1bi 1145	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑂 → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
26	25	eqcomd 2737	. . . . . . 7 ⊢ (𝑤 ∈ 𝑂 → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = 𝑤)
27	26	adantl 481	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = 𝑤)
28	23, 27	eqtr2d 2767	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
29	19, 22, 28	rspcedvd 3574	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
30	29	ralrimiva 3124	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
31		dffo3 7035	. . 3 ⊢ (𝐹:𝑃–onto→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)))
32	10, 30, 31	sylanbrc 583	. 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–onto→𝑂)
33		df-f1o 6488	. 2 ⊢ (𝐹:𝑃–1-1-onto→𝑂 ↔ (𝐹:𝑃–1-1→𝑂 ∧ 𝐹:𝑃–onto→𝑂))
34	18, 32, 33	sylanbrc 583	1 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395   ∩ cin 3896  𝒫 cpw 4547  {cpr 4575  ⟨cop 4579   class class class wbr 5089   ↦ cmpt 5170   Or wor 5521   × cxp 5612  ⟶wf 6477  –1-1→wf1 6478  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920  supcsup 9324  infcinf 9325  2c2 12180  ♯chash 14237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-hash 14238

This theorem is referenced by:  prproropen  47618  prproropreud  47619