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Theorem 1prl 7532
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 7444 . . 3 1P = ⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩
21fveq2i 5513 . 2 (1st ‘1P) = (1st ‘⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩)
3 ltnqex 7526 . . 3 {𝑥𝑥 <Q 1Q} ∈ V
4 gtnqex 7527 . . 3 {𝑦 ∣ 1Q <Q 𝑦} ∈ V
53, 4op1st 6140 . 2 (1st ‘⟨{𝑥𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}⟩) = {𝑥𝑥 <Q 1Q}
62, 5eqtri 2198 1 (1st ‘1P) = {𝑥𝑥 <Q 1Q}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  {cab 2163  cop 3594   class class class wbr 4000  cfv 5211  1st c1st 6132  1Qc1q 7258   <Q cltq 7262  1Pc1p 7269
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-iinf 4583
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-1st 6134  df-qs 6534  df-ni 7281  df-nqqs 7325  df-ltnqqs 7330  df-i1p 7444
This theorem is referenced by:  1idprl  7567  recexprlem1ssl  7610  recexprlemss1l  7612
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