Theorem 1pru 7913

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.

Step	Hyp	Ref	Expression
1		df-i1p 7824	. . 3 ⊢ 1P = ⟨{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩
2	1	fveq2i 5693	. 2 ⊢ (2nd ‘1P) = (2nd ‘⟨{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩)
3		ltnqex 7906	. . 3 ⊢ {𝑦 ∣ 𝑦 <Q 1Q} ∈ V
4		gtnqex 7907	. . 3 ⊢ {𝑥 ∣ 1Q <Q 𝑥} ∈ V
5	3, 4	op2nd 6371	. 2 ⊢ (2nd ‘⟨{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}⟩) = {𝑥 ∣ 1Q <Q 𝑥}
6	2, 5	eqtri 2259	1 ⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}

Syntax hints:   = wceq 1402  {cab 2224  ⟨cop 3708   class class class wbr 4125  ‘cfv 5372  2^nd c2nd 6363  1_Qc1q 7638   <_Q cltq 7642  1_Pc1p 7649

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-2nd 6365  df-qs 6803  df-ni 7661  df-nqqs 7705  df-ltnqqs 7710  df-i1p 7824

This theorem is referenced by:  1idpru  7948  recexprlem1ssu  7991  recexprlemss1u  7993