Theorem xpexcnvm 5142

Description: A condition where the converse of xpex 4891 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

Assertion

Ref	Expression
xpexcnvm	⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)

Distinct variable group:   𝑥,𝐵

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem xpexcnvm

Step	Hyp	Ref	Expression
1		dmexg 5046	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2		dmxpm 5002	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	eleq1d 2307	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
4	1, 3	imbitrid 154	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
5	4	imp 124	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   × cxp 4772  dom cdm 4774

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

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-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 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by: (None)