Theorem prproropf1o 47494

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 47491	. . . 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 9516	. . . . . . 7 ⊢ (𝑝 = 𝑤 → inf(𝑝, 𝑉, 𝑅) = inf(𝑤, 𝑉, 𝑅))
6		supeq1 9485	. . . . . . 7 ⊢ (𝑝 = 𝑤 → sup(𝑝, 𝑉, 𝑅) = sup(𝑤, 𝑉, 𝑅))
7	5, 6	opeq12d 4881	. . . . . 6 ⊢ (𝑝 = 𝑤 → ⟨inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)⟩ = ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
8	7	cbvmptv 5255	. . . . 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 2765	. . . 4 ⊢ 𝐹 = (𝑤 ∈ 𝑃 ↦ ⟨inf(𝑤, 𝑉, 𝑅), sup(𝑤, 𝑉, 𝑅)⟩)
10	3, 9	fmptd 7134	. . 3 ⊢ (𝑅 Or 𝑉 → 𝐹:𝑃⟶𝑂)
11		3ancomb 1099	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑃 ∧ 𝑧 ∈ 𝑃) ↔ (𝑅 Or 𝑉 ∧ 𝑧 ∈ 𝑃 ∧ 𝑤 ∈ 𝑃))
12		3anass 1095	. . . . . 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 47493	. . . . 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 3202	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑧 ∈ 𝑃 ∀𝑤 ∈ 𝑃 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
17		dff13 7275	. . 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 47490	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → {(1st ‘𝑤), (2nd ‘𝑤)} ∈ 𝑃)
20		fveq2 6906	. . . . . . 7 ⊢ (𝑧 = {(1st ‘𝑤), (2nd ‘𝑤)} → (𝐹‘𝑧) = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
21	20	eqeq2d 2748	. . . . . 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 47492	. . . . . 6 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}) = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
24	1	prproropf1olem0 47489	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑂 ↔ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∧ ((1st ‘𝑤) ∈ 𝑉 ∧ (2nd ‘𝑤) ∈ 𝑉) ∧ (1st ‘𝑤)𝑅(2nd ‘𝑤)))
25	24	simp1bi 1146	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑂 → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
26	25	eqcomd 2743	. . . . . . 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 2778	. . . . 5 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → 𝑤 = (𝐹‘{(1st ‘𝑤), (2nd ‘𝑤)}))
29	19, 22, 28	rspcedvd 3624	. . . 4 ⊢ ((𝑅 Or 𝑉 ∧ 𝑤 ∈ 𝑂) → ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
30	29	ralrimiva 3146	. . 3 ⊢ (𝑅 Or 𝑉 → ∀𝑤 ∈ 𝑂 ∃𝑧 ∈ 𝑃 𝑤 = (𝐹‘𝑧))
31		dffo3 7122	. . 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 6568	. 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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436   ∩ cin 3950  𝒫 cpw 4600  {cpr 4628  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225   Or wor 5591   × cxp 5683  ⟶wf 6557  –1-1→wf1 6558  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013  supcsup 9480  infcinf 9481  2c2 12321  ♯chash 14369

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370

This theorem is referenced by:  prproropen  47495  prproropreud  47496