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Theorem txuni2 22149
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 5546 . . 3 Rel (𝑋 × 𝑌)
2 txuni2.1 . . . . . . . 8 𝑋 = 𝑅
32eleq2i 2903 . . . . . . 7 (𝑧𝑋𝑧 𝑅)
4 eluni2 4815 . . . . . . 7 (𝑧 𝑅 ↔ ∃𝑟𝑅 𝑧𝑟)
53, 4bitri 278 . . . . . 6 (𝑧𝑋 ↔ ∃𝑟𝑅 𝑧𝑟)
6 txuni2.2 . . . . . . . 8 𝑌 = 𝑆
76eleq2i 2903 . . . . . . 7 (𝑤𝑌𝑤 𝑆)
8 eluni2 4815 . . . . . . 7 (𝑤 𝑆 ↔ ∃𝑠𝑆 𝑤𝑠)
97, 8bitri 278 . . . . . 6 (𝑤𝑌 ↔ ∃𝑠𝑆 𝑤𝑠)
105, 9anbi12i 629 . . . . 5 ((𝑧𝑋𝑤𝑌) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
11 opelxp 5564 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧𝑋𝑤𝑌))
12 reeanv 3352 . . . . 5 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) ↔ (∃𝑟𝑅 𝑧𝑟 ∧ ∃𝑠𝑆 𝑤𝑠))
1310, 11, 123bitr4i 306 . . . 4 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠))
14 opelxp 5564 . . . . . 6 (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧𝑟𝑤𝑠))
15 eqid 2821 . . . . . . . . . 10 (𝑟 × 𝑠) = (𝑟 × 𝑠)
16 xpeq1 5542 . . . . . . . . . . . 12 (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
1716eqeq2d 2832 . . . . . . . . . . 11 (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18 xpeq2 5549 . . . . . . . . . . . 12 (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
1918eqeq2d 2832 . . . . . . . . . . 11 (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
2017, 19rspc2ev 3612 . . . . . . . . . 10 ((𝑟𝑅𝑠𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
2115, 20mp3an3 1447 . . . . . . . . 9 ((𝑟𝑅𝑠𝑆) → ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22 vex 3474 . . . . . . . . . . 11 𝑟 ∈ V
23 vex 3474 . . . . . . . . . . 11 𝑠 ∈ V
2422, 23xpex 7451 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
25 eqeq1 2825 . . . . . . . . . . 11 (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26252rexbidv 3286 . . . . . . . . . 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 2821 . . . . . . . . . . . 12 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
2928rnmpo 7258 . . . . . . . . . . 11 ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3027, 29eqtri 2844 . . . . . . . . . 10 𝐵 = {𝑧 ∣ ∃𝑥𝑅𝑦𝑆 𝑧 = (𝑥 × 𝑦)}
3124, 26, 30elab2 3647 . . . . . . . . 9 ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥𝑅𝑦𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
3221, 31sylibr 237 . . . . . . . 8 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33 elssuni 4841 . . . . . . . 8 ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ 𝐵)
3432, 33syl 17 . . . . . . 7 ((𝑟𝑅𝑠𝑆) → (𝑟 × 𝑠) ⊆ 𝐵)
3534sseld 3942 . . . . . 6 ((𝑟𝑅𝑠𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3614, 35syl5bir 246 . . . . 5 ((𝑟𝑅𝑠𝑆) → ((𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵))
3736rexlimivv 3278 . . . 4 (∃𝑟𝑅𝑠𝑆 (𝑧𝑟𝑤𝑠) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
3813, 37sylbi 220 . . 3 (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ 𝐵)
391, 38relssi 5633 . 2 (𝑋 × 𝑌) ⊆ 𝐵
40 elssuni 4841 . . . . . . . . . 10 (𝑥𝑅𝑥 𝑅)
4140, 2sseqtrrdi 3994 . . . . . . . . 9 (𝑥𝑅𝑥𝑋)
42 elssuni 4841 . . . . . . . . . 10 (𝑦𝑆𝑦 𝑆)
4342, 6sseqtrrdi 3994 . . . . . . . . 9 (𝑦𝑆𝑦𝑌)
44 xpss12 5543 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
4541, 43, 44syl2an 598 . . . . . . . 8 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46 vex 3474 . . . . . . . . . 10 𝑥 ∈ V
47 vex 3474 . . . . . . . . . 10 𝑦 ∈ V
4846, 47xpex 7451 . . . . . . . . 9 (𝑥 × 𝑦) ∈ V
4948elpw 4516 . . . . . . . 8 ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
5045, 49sylibr 237 . . . . . . 7 ((𝑥𝑅𝑦𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
5150rgen2 3191 . . . . . 6 𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
5228fmpo 7741 . . . . . 6 (∀𝑥𝑅𝑦𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌))
5351, 52mpbi 233 . . . . 5 (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54 frn 6493 . . . . 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 3977 . . 3 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57 sspwuni 4995 . . 3 (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ 𝐵 ⊆ (𝑋 × 𝑌))
5856, 57mpbi 233 . 2 𝐵 ⊆ (𝑋 × 𝑌)
5939, 58eqssi 3959 1 (𝑋 × 𝑌) = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  {cab 2799  wral 3126  wrex 3127  wss 3910  𝒫 cpw 4512  cop 4546   cuni 4811   × cxp 5526  ran crn 5529  wf 6324  cmpo 7132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665
This theorem is referenced by:  txbasex  22150  txtopon  22175  sxsigon  31459
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