Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropreud | Structured version Visualization version GIF version |
Description: There is exactly one ordered ordered pair fulfilling a wff iff there is exactly one proper pair fulfilling an equivalent wff. (Contributed by AV, 20-Mar-2023.) |
Ref | Expression |
---|---|
prproropreud.o | ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) |
prproropreud.p | ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} |
prproropreud.b | ⊢ (𝜑 → 𝑅 Or 𝑉) |
prproropreud.x | ⊢ (𝑥 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 → (𝜓 ↔ 𝜒)) |
prproropreud.z | ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
prproropreud | ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prproropreud.b | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝑉) | |
2 | prproropreud.o | . . . . 5 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) | |
3 | prproropreud.p | . . . . 5 ⊢ 𝑃 = {𝑝 ∈ 𝒫 𝑉 ∣ (♯‘𝑝) = 2} | |
4 | eqid 2821 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) | |
5 | 2, 3, 4 | prproropf1o 43718 | . . . 4 ⊢ (𝑅 Or 𝑉 → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉):𝑃–1-1-onto→𝑂) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉):𝑃–1-1-onto→𝑂) |
7 | sbceq1a 3783 | . . . 4 ⊢ (𝑥 = ((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) → (𝜓 ↔ [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) | |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = ((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦)) → (𝜓 ↔ [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) |
9 | prproropreud.z | . . 3 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜃)) | |
10 | nfsbc1v 3792 | . . 3 ⊢ Ⅎ𝑥[((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 | |
11 | 6, 8, 9, 10 | reuf1odnf 43355 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) |
12 | eqidd 2822 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉) = (𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)) | |
13 | infeq1 8940 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → inf(𝑝, 𝑉, 𝑅) = inf(𝑦, 𝑉, 𝑅)) | |
14 | supeq1 8909 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → sup(𝑝, 𝑉, 𝑅) = sup(𝑦, 𝑉, 𝑅)) | |
15 | 13, 14 | opeq12d 4811 | . . . . . . 7 ⊢ (𝑝 = 𝑦 → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
16 | 15 | adantl 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑝 = 𝑦) → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
17 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝑦 ∈ 𝑃) | |
18 | opex 5356 | . . . . . . 7 ⊢ 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 ∈ V | |
19 | 18 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 ∈ V) |
20 | 12, 16, 17, 19 | fvmptd 6775 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
21 | 20 | sbceq1d 3777 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ([((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 ↔ [〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 / 𝑥]𝜓)) |
22 | prproropreud.x | . . . . . 6 ⊢ (𝑥 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 → (𝜓 ↔ 𝜒)) | |
23 | 22 | sbcieg 3810 | . . . . 5 ⊢ (〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 ∈ V → ([〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 / 𝑥]𝜓 ↔ 𝜒)) |
24 | 19, 23 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ([〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉 / 𝑥]𝜓 ↔ 𝜒)) |
25 | 21, 24 | bitrd 281 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ([((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 ↔ 𝜒)) |
26 | 25 | reubidva 3388 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) |
27 | 11, 26 | bitrd 281 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!wreu 3140 {crab 3142 Vcvv 3494 [wsbc 3772 ∩ cin 3935 𝒫 cpw 4539 〈cop 4573 ↦ cmpt 5146 Or wor 5473 × cxp 5553 –1-1-onto→wf1o 6354 ‘cfv 6355 supcsup 8904 infcinf 8905 2c2 11693 ♯chash 13691 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: (None) |
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