Theorem prproropf1o 48145

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 48142	. . . 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 9437	. . . . . . 7 ⊢ (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅))
6		supeq1 9405	. . . . . . 7 ⊢ (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅))
7	5, 6	opeq12d 4850	. . . . . 6 ⊢ (𝑝 = 𝑤 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
8	7	cbvmptv 5219	. . . . 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 2792	. . . 4 ⊢ 𝐹 = (𝑤 ∈ 𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
10	3, 9	fmptd 7110	. . 3 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃⟶𝑂)
11		3ancomb 1114	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))
12		3anass 1109	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)))
13	11, 12	bitri 278	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)))
14	1, 2, 4	prproropf1olem4 48144	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
15	13, 14	sylbir 238	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ (𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
16	15	ralrimivva 3214	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
17		dff13 7253	. . 3 ⊢ (𝐹:𝑃–1-1→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
18	10, 16, 17	sylanbrc 594	. 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1→𝑂)
19	1, 2	prproropf1olem1 48141	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → {(1st ‘𝑤), (2nd ‘𝑤)} ∈ 𝑃)
20		fveq2 6882	. . . . . . 7 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝐹‘𝑧) = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
21	20	eqeq2d 2780	. . . . . 6 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)})))
22	21	adantl 486	. . . . 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 48143	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}) = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
24	1	prproropf1olem0 48140	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑂 ↔ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∧ ((1st ‘𝑤) ∈ 𝑉 ∧ (2nd ‘𝑤) ∈ 𝑉) ∧ (1st ‘𝑤)𝑅(2nd ‘𝑤)))
25	24	simp1bi 1161	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑂 → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
26	25	eqcomd 2775	. . . . . . 7 ⊢ (𝑤 ∈ 𝑂 → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = 𝑤)
27	26	adantl 486	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = 𝑤)
28	23, 27	eqtr2d 2805	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
29	19, 22, 28	rspcedvd 3592	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
30	29	ralrimiva 3163	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
31		dffo3 7098	. . 3 ⊢ (𝐹:𝑃–onto→𝑂 ↔ (𝐹:𝑃⟶𝑂 ∧ ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧)))
32	10, 30, 31	sylanbrc 594	. 2 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–onto→𝑂)
33		df-f1o 6544	. 2 ⊢ (𝐹:𝑃–1-1-onto→𝑂 ↔ (𝐹:𝑃–1-1→𝑂 ∧ 𝐹:𝑃–onto→𝑂))
34	18, 32, 33	sylanbrc 594	1 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃–1-1-onto→𝑂)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423   ∩ cin 3912  𝒫 cpw 4567  {cpr 4596  ⟨cop 4600   class class class wbr 5113   ↦ cmpt 5196   Or wor 5569   × cxp 5660  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  1^st c1st 7984  2^nd c2nd 7985  supcsup 9400  infcinf 9401  2c2 12295  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-hash 14367

This theorem is referenced by:  prproropen  48146  prproropreud  48147