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Mirrors > Home > ILE Home > Th. List > xpexgALT | Unicode version |
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4552 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
xpexgALT |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 3785 |
. . . 4
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2 | 1 | xpeq2i 4459 |
. . 3
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3 | xpiundi 4496 |
. . 3
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4 | 2, 3 | eqtr3i 2110 |
. 2
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5 | id 19 |
. . 3
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6 | fconstmpt 4485 |
. . . . 5
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7 | mptexg 5522 |
. . . . 5
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8 | 6, 7 | syl5eqel 2174 |
. . . 4
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9 | 8 | ralrimivw 2447 |
. . 3
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10 | iunexg 5890 |
. . 3
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11 | 5, 9, 10 | syl2anr 284 |
. 2
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12 | 4, 11 | syl5eqel 2174 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 |
This theorem is referenced by: (None) |
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