Theorem djuf1olemr 7070

Description: Lemma for djulf1or 7072 and djurf1or 7073. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7069. (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 4917	. . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4		ssv 3191	. . . 4 ⊢ 𝐴 ⊆ V
5		resmpt 4969	. . . 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 2209	. 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)
8	1, 7	djuf1olem 7069	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   = wceq 1363   ∈ wcel 2159  Vcvv 2751   ⊆ wss 3143  {csn 3606  ⟨cop 3609   ↦ cmpt 4078   × cxp 4638   ↾ cres 4642  –1-1-onto→wf1o 5229

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-1st 6158  df-2nd 6159

This theorem is referenced by:  djulf1or  7072  djurf1or  7073