Theorem djuf1olemr 7400

Description: Lemma for djulf1or 7402 and djurf1or 7403. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7399. (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 5061	. . 3 ⊢ (𝐹 ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩) ↾ 𝐴)
4		ssv 3270	. . . 4 ⊢ 𝐴 ⊆ V
5		resmpt 5114	. . . 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 2259	. 2 ⊢ (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)
8	1, 7	djuf1olem 7399	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {csn 3709  ⟨cop 3712   ↦ cmpt 4193   × cxp 4774   ↾ cres 4778  –1-1-onto→wf1o 5379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379

This theorem is used by:  djulf1or  7402  djurf1or  7403