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Theorem djuf1olemr 6744
Description: Lemma for djulf1or 6746 and djurf1or 6747. Remark: maybe a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion would be more usable. (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 4709 . . 3 (𝐹𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4 ssv 3046 . . . 4 𝐴 ⊆ V
5 resmpt 4760 . . . 4 (𝐴 ⊆ V → ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩))
64, 5ax-mp 7 . . 3 ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
73, 6eqtri 2108 . 2 (𝐹𝐴) = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
81, 7djuf1olem 6743 1 (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  Vcvv 2619  wss 2999  {csn 3446  cop 3449  cmpt 3899   × cxp 4436  cres 4440  1-1-ontowf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  djulf1or  6746  djurf1or  6747
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