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Mirrors > Home > ILE Home > Th. List > suplocexprlem2b | GIF version |
Description: Lemma for suplocexpr 7533. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Ref | Expression |
---|---|
suplocexprlem2b.b | ⊢ 𝐵 = 〈∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}〉 |
Ref | Expression |
---|---|
suplocexprlem2b | ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suplocexprlem2b.b | . . 3 ⊢ 𝐵 = 〈∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}〉 | |
2 | 1 | fveq2i 5424 | . 2 ⊢ (2nd ‘𝐵) = (2nd ‘〈∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}〉) |
3 | fo1st 6055 | . . . . . 6 ⊢ 1st :V–onto→V | |
4 | fofun 5346 | . . . . . 6 ⊢ (1st :V–onto→V → Fun 1st ) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ Fun 1st |
6 | npex 7281 | . . . . . 6 ⊢ P ∈ V | |
7 | 6 | ssex 4065 | . . . . 5 ⊢ (𝐴 ⊆ P → 𝐴 ∈ V) |
8 | funimaexg 5207 | . . . . 5 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V) | |
9 | 5, 7, 8 | sylancr 410 | . . . 4 ⊢ (𝐴 ⊆ P → (1st “ 𝐴) ∈ V) |
10 | uniexg 4361 | . . . 4 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝐴 ⊆ P → ∪ (1st “ 𝐴) ∈ V) |
12 | nqex 7171 | . . . 4 ⊢ Q ∈ V | |
13 | 12 | rabex 4072 | . . 3 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V |
14 | op2ndg 6049 | . . 3 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (2nd ‘〈∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}〉) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}) | |
15 | 11, 13, 14 | sylancl 409 | . 2 ⊢ (𝐴 ⊆ P → (2nd ‘〈∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}〉) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}) |
16 | 2, 15 | syl5eq 2184 | 1 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 {crab 2420 Vcvv 2686 ⊆ wss 3071 〈cop 3530 ∪ cuni 3736 ∩ cint 3771 class class class wbr 3929 “ cima 4542 Fun wfun 5117 –onto→wfo 5121 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 Qcnq 7088 <Q cltq 7093 Pcnp 7099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-qs 6435 df-ni 7112 df-nqqs 7156 df-inp 7274 |
This theorem is referenced by: suplocexprlemmu 7526 suplocexprlemru 7527 suplocexprlemdisj 7528 suplocexprlemloc 7529 suplocexprlemex 7530 suplocexprlemub 7531 |
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