Theorem xpexcnvm 5122

Description: A condition where the converse of xpex 4871 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 5026	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2		dmxpm 4982	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	eleq1d 2303	. . 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)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1541   ∈ wcel 2205  Vcvv 2815   × cxp 4752  dom cdm 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by: (None)