Theorem djuf1olemr 7115

Description: Lemma for djulf1or 7117 and djurf1or 7118. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7114. (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	⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)

Distinct variable groups:   𝑥,𝑋   𝑥,𝐴

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem djuf1olemr

Step	Hyp	Ref	Expression
1		djuf1olemr.1	. 2 ⊢ 𝑋 ∈ V
2		djuf1olemr.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)
3	2	reseq1i 4939	. . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4		ssv 3202	. . . 4 ⊢ 𝐴 ⊆ V
5		resmpt 4991	. . . 4 ⊢ (𝐴 ⊆ V → ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩))
6	4, 5	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)
7	3, 6	eqtri 2214	. 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)
8	1, 7	djuf1olem 7114	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154  {csn 3619  ⟨cop 3622   ↦ cmpt 4091   × cxp 4658   ↾ cres 4662  –1-1-onto→wf1o 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by:  djulf1or  7117  djurf1or  7118