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Theorem 1pru 7478
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 7389 . . 3 1P = ⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩
21fveq2i 5473 . 2 (2nd ‘1P) = (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩)
3 ltnqex 7471 . . 3 {𝑦𝑦 <Q 1Q} ∈ V
4 gtnqex 7472 . . 3 {𝑥 ∣ 1Q <Q 𝑥} ∈ V
53, 4op2nd 6097 . 2 (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩) = {𝑥 ∣ 1Q <Q 𝑥}
62, 5eqtri 2178 1 (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Colors of variables: wff set class
Syntax hints:   = wceq 1335  {cab 2143  cop 3564   class class class wbr 3967  cfv 5172  2nd c2nd 6089  1Qc1q 7203   <Q cltq 7207  1Pc1p 7214
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-2nd 6091  df-qs 6488  df-ni 7226  df-nqqs 7270  df-ltnqqs 7275  df-i1p 7389
This theorem is referenced by:  1idpru  7513  recexprlem1ssu  7556  recexprlemss1u  7558
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