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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 2737 | . . . . 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 47494 | . . . 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 3799 | . . . 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 481 | . . 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 3808 | . . 3 ⊢ Ⅎ𝑥[((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 | |
| 11 | 6, 8, 9, 10 | reuf1odnf 47119 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) |
| 12 | eqidd 2738 | . . . . . 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 9516 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → inf(𝑝, 𝑉, 𝑅) = inf(𝑦, 𝑉, 𝑅)) | |
| 14 | supeq1 9485 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → sup(𝑝, 𝑉, 𝑅) = sup(𝑦, 𝑉, 𝑅)) | |
| 15 | 13, 14 | opeq12d 4881 | . . . . . . 7 ⊢ (𝑝 = 𝑦 → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
| 16 | 15 | adantl 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑝 = 𝑦) → 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉 = 〈inf(𝑦, 𝑉, 𝑅), sup(𝑦, 𝑉, 𝑅)〉) |
| 17 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝑦 ∈ 𝑃) | |
| 18 | opex 5469 | . . . . . . 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 7023 | . . . . 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 3793 | . . . 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 3828 | . . . . 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 3396 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!wreu 3378 {crab 3436 Vcvv 3480 [wsbc 3788 ∩ cin 3950 𝒫 cpw 4600 〈cop 4632 ↦ cmpt 5225 Or wor 5591 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 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: (None) |
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