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Theorem djuf1olemr 6939
 Description: Lemma for djulf1or 6941 and djurf1or 6942. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6938. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
Hypotheses
Ref Expression
djuf1olemr.1 𝑋 ∈ V
djuf1olemr.2 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
Assertion
Ref Expression
djuf1olemr (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)
Distinct variable groups:   𝑥,𝑋   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem djuf1olemr
StepHypRef Expression
1 djuf1olemr.1 . 2 𝑋 ∈ V
2 djuf1olemr.2 . . . 4 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
32reseq1i 4815 . . 3 (𝐹𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4 ssv 3119 . . . 4 𝐴 ⊆ V
5 resmpt 4867 . . . 4 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩))
64, 5ax-mp 5 . . 3 ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
73, 6eqtri 2160 . 2 (𝐹𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
81, 7djuf1olem 6938 1 (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  {csn 3527  ⟨cop 3530   ↦ cmpt 3989   × cxp 4537   ↾ cres 4541  –1-1-onto→wf1o 5122 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039 This theorem is referenced by:  djulf1or  6941  djurf1or  6942
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