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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 2735 | . . . . 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 47432 | . . . 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 3802 | . . . 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 3811 | . . 3 ⊢ Ⅎ𝑥[((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓 | |
11 | 6, 8, 9, 10 | reuf1odnf 47057 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝑂 𝜓 ↔ ∃!𝑦 ∈ 𝑃 [((𝑝 ∈ 𝑃 ↦ 〈inf(𝑝, 𝑉, 𝑅), sup(𝑝, 𝑉, 𝑅)〉)‘𝑦) / 𝑥]𝜓)) |
12 | eqidd 2736 | . . . . . 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 9514 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → inf(𝑝, 𝑉, 𝑅) = inf(𝑦, 𝑉, 𝑅)) | |
14 | supeq1 9483 | . . . . . . . 8 ⊢ (𝑝 = 𝑦 → sup(𝑝, 𝑉, 𝑅) = sup(𝑦, 𝑉, 𝑅)) | |
15 | 13, 14 | opeq12d 4886 | . . . . . . 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 5475 | . . . . . . 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 3796 | . . . 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 3832 | . . . . 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 3394 | . 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 1537 ∈ wcel 2106 ∃!wreu 3376 {crab 3433 Vcvv 3478 [wsbc 3791 ∩ cin 3962 𝒫 cpw 4605 〈cop 4637 ↦ cmpt 5231 Or wor 5596 × cxp 5687 –1-1-onto→wf1o 6562 ‘cfv 6563 supcsup 9478 infcinf 9479 2c2 12319 ♯chash 14366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: (None) |
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