Theorem xpexcnvm 5098

Description: A condition where the converse of xpex 4848 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 5002	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2		dmxpm 4958	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	eleq1d 2300	. . 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 2202  Vcvv 2803   × cxp 4729  dom cdm 4731

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by: (None)