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Theorem 1pru 7115
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 7026 . . 3 1P = ⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩
21fveq2i 5308 . 2 (2nd ‘1P) = (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩)
3 ltnqex 7108 . . 3 {𝑦𝑦 <Q 1Q} ∈ V
4 gtnqex 7109 . . 3 {𝑥 ∣ 1Q <Q 𝑥} ∈ V
53, 4op2nd 5918 . 2 (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩) = {𝑥 ∣ 1Q <Q 𝑥}
62, 5eqtri 2108 1 (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  {cab 2074  cop 3449   class class class wbr 3845  cfv 5015  2nd c2nd 5910  1Qc1q 6840   <Q cltq 6844  1Pc1p 6851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-2nd 5912  df-qs 6298  df-ni 6863  df-nqqs 6907  df-ltnqqs 6912  df-i1p 7026
This theorem is referenced by:  1idpru  7150  recexprlem1ssu  7193  recexprlemss1u  7195
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