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Theorem 1pru 7554
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 7465 . . 3 1P = ⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩
21fveq2i 5518 . 2 (2nd ‘1P) = (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩)
3 ltnqex 7547 . . 3 {𝑦𝑦 <Q 1Q} ∈ V
4 gtnqex 7548 . . 3 {𝑥 ∣ 1Q <Q 𝑥} ∈ V
53, 4op2nd 6147 . 2 (2nd ‘⟨{𝑦𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩) = {𝑥 ∣ 1Q <Q 𝑥}
62, 5eqtri 2198 1 (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  {cab 2163  cop 3595   class class class wbr 4003  cfv 5216  2nd c2nd 6139  1Qc1q 7279   <Q cltq 7283  1Pc1p 7290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-2nd 6141  df-qs 6540  df-ni 7302  df-nqqs 7346  df-ltnqqs 7351  df-i1p 7465
This theorem is referenced by:  1idpru  7589  recexprlem1ssu  7632  recexprlemss1u  7634
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