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Theorem 1pru 7357
Description: The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
Assertion
Ref Expression
1pru (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}

Proof of Theorem 1pru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-i1p 7268 . . 3 1P = ⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩
21fveq2i 5417 . 2 (2nd ‘1P) = (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩)
3 ltnqex 7350 . . 3 {𝑦𝑦 <Q 1Q} ∈ V
4 gtnqex 7351 . . 3 {𝑥 ∣ 1Q <Q 𝑥} ∈ V
53, 4op2nd 6038 . 2 (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩) = {𝑥 ∣ 1Q <Q 𝑥}
62, 5eqtri 2158 1 (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Colors of variables: wff set class
Syntax hints:   = wceq 1331  {cab 2123  cop 3525   class class class wbr 3924  cfv 5118  2nd c2nd 6030  1Qc1q 7082   <Q cltq 7086  1Pc1p 7093
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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-ltnqqs 7154  df-i1p 7268
This theorem is referenced by:  1idpru  7392  recexprlem1ssu  7435  recexprlemss1u  7437
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