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Mirrors > Home > ILE Home > Th. List > rnxpm | GIF version |
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
rnxpm | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 4410 | . . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 4802 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 4593 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | 1, 3 | eqtri 2103 | . 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
5 | dmxpm 4612 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → dom (𝐵 × 𝐴) = 𝐵) | |
6 | 4, 5 | syl5eq 2127 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∃wex 1422 ∈ wcel 1434 × cxp 4397 ◡ccnv 4398 dom cdm 4399 ran crn 4400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-xp 4405 df-rel 4406 df-cnv 4407 df-dm 4409 df-rn 4410 |
This theorem is referenced by: ssxpbm 4818 ssxp2 4820 xpexr2m 4824 xpima2m 4830 unixpm 4918 exmidfodomrlemim 6728 |
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