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Theorem 1prl 7386
Description: The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
Assertion
Ref Expression
1prl (1st ‘1P) = {𝑥𝑥 <Q 1Q}

Proof of Theorem 1prl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-i1p 7298 . . 3 1P = ⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩
21fveq2i 5431 . 2 (1st ‘1P) = (1st ‘⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩)
3 ltnqex 7380 . . 3 {𝑥𝑥 <Q 1Q} ∈ V
4 gtnqex 7381 . . 3 {𝑦 ∣ 1Q <Q 𝑦} ∈ V
53, 4op1st 6051 . 2 (1st ‘⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩) = {𝑥𝑥 <Q 1Q}
62, 5eqtri 2161 1 (1st ‘1P) = {𝑥𝑥 <Q 1Q}
Colors of variables: wff set class
Syntax hints:   = wceq 1332  {cab 2126  cop 3534   class class class wbr 3936  cfv 5130  1st c1st 6043  1Qc1q 7112   <Q cltq 7116  1Pc1p 7123
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-qs 6442  df-ni 7135  df-nqqs 7179  df-ltnqqs 7184  df-i1p 7298
This theorem is referenced by:  1idprl  7421  recexprlem1ssl  7464  recexprlemss1l  7466
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