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Theorem djuf1olemr 7043
Description: Lemma for djulf1or 7045 and djurf1or 7046. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7042. (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 4896 . . 3 (𝐹𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4 ssv 3175 . . . 4 𝐴 ⊆ V
5 resmpt 4948 . . . 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 2196 . 2 (𝐹𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
81, 7djuf1olem 7042 1 (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2146  Vcvv 2735  wss 3127  {csn 3589  cop 3592  cmpt 4059   × cxp 4618  cres 4622  1-1-ontowf1o 5207
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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132
This theorem is referenced by:  djulf1or  7045  djurf1or  7046
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