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Theorem txuni2 13841
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 4737 . . 3 Rel (𝑋 × 𝑌)
2 txuni2.1 . . . . . . . 8 𝑋 = 𝑅
32eleq2i 2244 . . . . . . 7 (𝑧𝑋𝑧 𝑅)
4 eluni2 3815 . . . . . . 7 (𝑧 𝑅 ↔ ∃𝑟𝑅 𝑧𝑟)
53, 4bitri 184 . . . . . 6 (𝑧𝑋 ↔ ∃𝑟𝑅 𝑧𝑟)
6 txuni2.2 . . . . . . . 8 𝑌 = 𝑆
76eleq2i 2244 . . . . . . 7 (𝑤𝑌𝑤 𝑆)
8 eluni2 3815 . . . . . . 7 (𝑤 𝑆 ↔ ∃𝑠𝑆 𝑤𝑠)
97, 8bitri 184 . . . . . 6 (𝑤𝑌 ↔ ∃𝑠𝑆 𝑤𝑠)
105, 9anbi12i 460 . . . . 5 ((𝑧𝑋𝑤𝑌) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
11 opelxp 4658 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧𝑋𝑤𝑌))
12 reeanv 2647 . . . . 5 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
1310, 11, 123bitr4i 212 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠))
14 opelxp 4658 . . . . . 6 (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧𝑟𝑤𝑠))
15 eqid 2177 . . . . . . . . . 10 (𝑟 × 𝑠) = (𝑟 × 𝑠)
16 xpeq1 4642 . . . . . . . . . . . 12 (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
1716eqeq2d 2189 . . . . . . . . . . 11 (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18 xpeq2 4643 . . . . . . . . . . . 12 (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
1918eqeq2d 2189 . . . . . . . . . . 11 (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
2017, 19rspc2ev 2858 . . . . . . . . . 10 ((𝑟𝑅𝑠𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
2115, 20mp3an3 1326 . . . . . . . . 9 ((𝑟𝑅𝑠𝑆) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22 vex 2742 . . . . . . . . . . 11 𝑟 ∈ V
23 vex 2742 . . . . . . . . . . 11 𝑠 ∈ V
2422, 23xpex 4743 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
25 eqeq1 2184 . . . . . . . . . . 11 (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26252rexbidv 2502 . . . . . . . . . 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 2177 . . . . . . . . . . . 12 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
2928rnmpo 5987 . . . . . . . . . . 11 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3027, 29eqtri 2198 . . . . . . . . . 10 𝐵 = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3124, 26, 30elab2 2887 . . . . . . . . 9 ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
3221, 31sylibr 134 . . . . . . . 8 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33 elssuni 3839 . . . . . . . 8 ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ 𝐵)
3432, 33syl 14 . . . . . . 7 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ⊆ 𝐵)
3534sseld 3156 . . . . . 6 ((𝑟𝑅𝑠𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3614, 35biimtrrid 153 . . . . 5 ((𝑟𝑅𝑠𝑆) → ((𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3736rexlimivv 2600 . . . 4 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
3813, 37sylbi 121 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
391, 38relssi 4719 . 2 (𝑋 × 𝑌) ⊆ 𝐵
40 elssuni 3839 . . . . . . . . . 10 (𝑥𝑅𝑥 𝑅)
4140, 2sseqtrrdi 3206 . . . . . . . . 9 (𝑥𝑅𝑥𝑋)
42 elssuni 3839 . . . . . . . . . 10 (𝑦𝑆𝑦 𝑆)
4342, 6sseqtrrdi 3206 . . . . . . . . 9 (𝑦𝑆𝑦𝑌)
44 xpss12 4735 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
4541, 43, 44syl2an 289 . . . . . . . 8 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46 vex 2742 . . . . . . . . . 10 𝑥 ∈ V
47 vex 2742 . . . . . . . . . 10 𝑦 ∈ V
4846, 47xpex 4743 . . . . . . . . 9 (𝑥 × 𝑦) ∈ V
4948elpw 3583 . . . . . . . 8 ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
5045, 49sylibr 134 . . . . . . 7 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
5150rgen2 2563 . . . . . 6 𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
5228fmpo 6204 . . . . . 6 (∀𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌))
5351, 52mpbi 145 . . . . 5 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54 frn 5376 . . . . 5 ((𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌) → ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌))
5553, 54ax-mp 5 . . . 4 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌)
5627, 55eqsstri 3189 . . 3 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57 sspwuni 3973 . . 3 (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ 𝐵 ⊆ (𝑋 × 𝑌))
5856, 57mpbi 145 . 2 𝐵 ⊆ (𝑋 × 𝑌)
5939, 58eqssi 3173 1 (𝑋 × 𝑌) = 𝐵
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  wss 3131  𝒫 cpw 3577  cop 3597   cuni 3811   × cxp 4626  ran crn 4629  wf 5214  cmpo 5879
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144
This theorem is referenced by:  txbasex  13842  txtopon  13847
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