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Mirrors > Home > MPE Home > Th. List > xpexcnv | Structured version Visualization version GIF version |
Description: A condition where the converse of xpex 7465 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.) |
Ref | Expression |
---|---|
xpexcnv | ⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7602 | . . 3 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V) | |
2 | dmxp 5792 | . . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
3 | 2 | eleq1d 2894 | . . 3 ⊢ (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V)) |
4 | 1, 3 | syl5ib 245 | . 2 ⊢ (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V)) |
5 | 4 | imp 407 | 1 ⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 × cxp 5546 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: fczsupp0 7848 |
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