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Theorem txuni2 12267
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
txval.1 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
txuni2.1 𝑋 = 𝑅
txuni2.2 𝑌 = 𝑆
Assertion
Ref Expression
txuni2 (𝑋 × 𝑌) = 𝐵
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem txuni2
Dummy variables 𝑟 𝑠 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4608 . . 3 Rel (𝑋 × 𝑌)
2 txuni2.1 . . . . . . . 8 𝑋 = 𝑅
32eleq2i 2181 . . . . . . 7 (𝑧𝑋𝑧 𝑅)
4 eluni2 3706 . . . . . . 7 (𝑧 𝑅 ↔ ∃𝑟𝑅 𝑧𝑟)
53, 4bitri 183 . . . . . 6 (𝑧𝑋 ↔ ∃𝑟𝑅 𝑧𝑟)
6 txuni2.2 . . . . . . . 8 𝑌 = 𝑆
76eleq2i 2181 . . . . . . 7 (𝑤𝑌𝑤 𝑆)
8 eluni2 3706 . . . . . . 7 (𝑤 𝑆 ↔ ∃𝑠𝑆 𝑤𝑠)
97, 8bitri 183 . . . . . 6 (𝑤𝑌 ↔ ∃𝑠𝑆 𝑤𝑠)
105, 9anbi12i 453 . . . . 5 ((𝑧𝑋𝑤𝑌) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
11 opelxp 4529 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧𝑋𝑤𝑌))
12 reeanv 2574 . . . . 5 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
1310, 11, 123bitr4i 211 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠))
14 opelxp 4529 . . . . . 6 (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧𝑟𝑤𝑠))
15 eqid 2115 . . . . . . . . . 10 (𝑟 × 𝑠) = (𝑟 × 𝑠)
16 xpeq1 4513 . . . . . . . . . . . 12 (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
1716eqeq2d 2126 . . . . . . . . . . 11 (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18 xpeq2 4514 . . . . . . . . . . . 12 (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
1918eqeq2d 2126 . . . . . . . . . . 11 (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
2017, 19rspc2ev 2774 . . . . . . . . . 10 ((𝑟𝑅𝑠𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
2115, 20mp3an3 1287 . . . . . . . . 9 ((𝑟𝑅𝑠𝑆) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22 vex 2660 . . . . . . . . . . 11 𝑟 ∈ V
23 vex 2660 . . . . . . . . . . 11 𝑠 ∈ V
2422, 23xpex 4614 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
25 eqeq1 2121 . . . . . . . . . . 11 (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26252rexbidv 2434 . . . . . . . . . 10 (𝑧 = (𝑟 × 𝑠) → (∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦) ↔ ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦)))
27 txval.1 . . . . . . . . . . 11 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
28 eqid 2115 . . . . . . . . . . . 12 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
2928rnmpo 5835 . . . . . . . . . . 11 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3027, 29eqtri 2135 . . . . . . . . . 10 𝐵 = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3124, 26, 30elab2 2801 . . . . . . . . 9 ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
3221, 31sylibr 133 . . . . . . . 8 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33 elssuni 3730 . . . . . . . 8 ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ 𝐵)
3432, 33syl 14 . . . . . . 7 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ⊆ 𝐵)
3534sseld 3062 . . . . . 6 ((𝑟𝑅𝑠𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3614, 35syl5bir 152 . . . . 5 ((𝑟𝑅𝑠𝑆) → ((𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3736rexlimivv 2529 . . . 4 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
3813, 37sylbi 120 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
391, 38relssi 4590 . 2 (𝑋 × 𝑌) ⊆ 𝐵
40 elssuni 3730 . . . . . . . . . 10 (𝑥𝑅𝑥 𝑅)
4140, 2syl6sseqr 3112 . . . . . . . . 9 (𝑥𝑅𝑥𝑋)
42 elssuni 3730 . . . . . . . . . 10 (𝑦𝑆𝑦 𝑆)
4342, 6syl6sseqr 3112 . . . . . . . . 9 (𝑦𝑆𝑦𝑌)
44 xpss12 4606 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
4541, 43, 44syl2an 285 . . . . . . . 8 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46 vex 2660 . . . . . . . . . 10 𝑥 ∈ V
47 vex 2660 . . . . . . . . . 10 𝑦 ∈ V
4846, 47xpex 4614 . . . . . . . . 9 (𝑥 × 𝑦) ∈ V
4948elpw 3482 . . . . . . . 8 ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
5045, 49sylibr 133 . . . . . . 7 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
5150rgen2 2492 . . . . . 6 𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
5228fmpo 6053 . . . . . 6 (∀𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌))
5351, 52mpbi 144 . . . . 5 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54 frn 5239 . . . . 5 ((𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌) → ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌))
5553, 54ax-mp 7 . . . 4 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌)
5627, 55eqsstri 3095 . . 3 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57 sspwuni 3863 . . 3 (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ 𝐵 ⊆ (𝑋 × 𝑌))
5856, 57mpbi 144 . 2 𝐵 ⊆ (𝑋 × 𝑌)
5939, 58eqssi 3079 1 (𝑋 × 𝑌) = 𝐵
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wcel 1463  {cab 2101  wral 2390  wrex 2391  wss 3037  𝒫 cpw 3476  cop 3496   cuni 3702   × cxp 4497  ran crn 4500  wf 5077  cmpo 5730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993
This theorem is referenced by:  txbasex  12268  txtopon  12273
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