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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 2733 | . . . . 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 46110 | . . . 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 3787 | . . . 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 483 | . . 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 3796 | . . 3 ⊢ Ⅎ𝑥[((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 | |
11 | 6, 8, 9, 10 | reuf1odnf 45750 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) |
12 | eqidd 2734 | . . . . . 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 9467 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → inf(𝑝, 𝑉, 𝑅) = inf(𝑦, 𝑉, 𝑅)) | |
14 | supeq1 9436 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → sup(𝑝, 𝑉, 𝑅) = sup(𝑦, 𝑉, 𝑅)) | |
15 | 13, 14 | opeq12d 4880 | . . . . . . 7 ⊢ (𝑝 = 𝑦 → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
16 | 15 | adantl 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑝 = 𝑦) → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
17 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝑦 ∈ 𝑃) | |
18 | opex 5463 | . . . . . . 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 7001 | . . . . 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 3781 | . . . 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 3816 | . . . . 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 279 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ([((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 ↔ 𝜒)) |
26 | 25 | reubidva 3393 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) |
27 | 11, 26 | bitrd 279 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃!wreu 3375 {crab 3433 Vcvv 3475 [wsbc 3776 ∩ cin 3946 𝒫 cpw 4601 〈cop 4633 ↦ cmpt 5230 Or wor 5586 × cxp 5673 –1-1-onto→wf1o 6539 ‘cfv 6540 supcsup 9431 infcinf 9432 2c2 12263 ♯chash 14286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 |
This theorem is referenced by: (None) |
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