Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  txuni2 GIF version

Theorem txuni2 12627
 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 4694 . . 3 Rel (𝑋 × 𝑌)
2 txuni2.1 . . . . . . . 8 𝑋 = 𝑅
32eleq2i 2224 . . . . . . 7 (𝑧𝑋𝑧 𝑅)
4 eluni2 3776 . . . . . . 7 (𝑧 𝑅 ↔ ∃𝑟𝑅 𝑧𝑟)
53, 4bitri 183 . . . . . 6 (𝑧𝑋 ↔ ∃𝑟𝑅 𝑧𝑟)
6 txuni2.2 . . . . . . . 8 𝑌 = 𝑆
76eleq2i 2224 . . . . . . 7 (𝑤𝑌𝑤 𝑆)
8 eluni2 3776 . . . . . . 7 (𝑤 𝑆 ↔ ∃𝑠𝑆 𝑤𝑠)
97, 8bitri 183 . . . . . 6 (𝑤𝑌 ↔ ∃𝑠𝑆 𝑤𝑠)
105, 9anbi12i 456 . . . . 5 ((𝑧𝑋𝑤𝑌) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
11 opelxp 4615 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧𝑋𝑤𝑌))
12 reeanv 2626 . . . . 5 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
1310, 11, 123bitr4i 211 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠))
14 opelxp 4615 . . . . . 6 (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧𝑟𝑤𝑠))
15 eqid 2157 . . . . . . . . . 10 (𝑟 × 𝑠) = (𝑟 × 𝑠)
16 xpeq1 4599 . . . . . . . . . . . 12 (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
1716eqeq2d 2169 . . . . . . . . . . 11 (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18 xpeq2 4600 . . . . . . . . . . . 12 (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
1918eqeq2d 2169 . . . . . . . . . . 11 (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
2017, 19rspc2ev 2831 . . . . . . . . . 10 ((𝑟𝑅𝑠𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
2115, 20mp3an3 1308 . . . . . . . . 9 ((𝑟𝑅𝑠𝑆) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22 vex 2715 . . . . . . . . . . 11 𝑟 ∈ V
23 vex 2715 . . . . . . . . . . 11 𝑠 ∈ V
2422, 23xpex 4700 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
25 eqeq1 2164 . . . . . . . . . . 11 (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26252rexbidv 2482 . . . . . . . . . 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 2157 . . . . . . . . . . . 12 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
2928rnmpo 5928 . . . . . . . . . . 11 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3027, 29eqtri 2178 . . . . . . . . . 10 𝐵 = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3124, 26, 30elab2 2860 . . . . . . . . 9 ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
3221, 31sylibr 133 . . . . . . . 8 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33 elssuni 3800 . . . . . . . 8 ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ 𝐵)
3432, 33syl 14 . . . . . . 7 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ⊆ 𝐵)
3534sseld 3127 . . . . . 6 ((𝑟𝑅𝑠𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3614, 35syl5bir 152 . . . . 5 ((𝑟𝑅𝑠𝑆) → ((𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3736rexlimivv 2580 . . . 4 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
3813, 37sylbi 120 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
391, 38relssi 4676 . 2 (𝑋 × 𝑌) ⊆ 𝐵
40 elssuni 3800 . . . . . . . . . 10 (𝑥𝑅𝑥 𝑅)
4140, 2sseqtrrdi 3177 . . . . . . . . 9 (𝑥𝑅𝑥𝑋)
42 elssuni 3800 . . . . . . . . . 10 (𝑦𝑆𝑦 𝑆)
4342, 6sseqtrrdi 3177 . . . . . . . . 9 (𝑦𝑆𝑦𝑌)
44 xpss12 4692 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
4541, 43, 44syl2an 287 . . . . . . . 8 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46 vex 2715 . . . . . . . . . 10 𝑥 ∈ V
47 vex 2715 . . . . . . . . . 10 𝑦 ∈ V
4846, 47xpex 4700 . . . . . . . . 9 (𝑥 × 𝑦) ∈ V
4948elpw 3549 . . . . . . . 8 ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
5045, 49sylibr 133 . . . . . . 7 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
5150rgen2 2543 . . . . . 6 𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
5228fmpo 6146 . . . . . 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 5327 . . . . 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 3160 . . 3 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57 sspwuni 3933 . . 3 (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ 𝐵 ⊆ (𝑋 × 𝑌))
5856, 57mpbi 144 . 2 𝐵 ⊆ (𝑋 × 𝑌)
5939, 58eqssi 3144 1 (𝑋 × 𝑌) = 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1335   ∈ wcel 2128  {cab 2143  ∀wral 2435  ∃wrex 2436   ⊆ wss 3102  𝒫 cpw 3543  ⟨cop 3563  ∪ cuni 3772   × cxp 4583  ran crn 4586  ⟶wf 5165   ∈ cmpo 5823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086 This theorem is referenced by:  txbasex  12628  txtopon  12633
 Copyright terms: Public domain W3C validator