Theorem txvalex 15278

Description: Existence of the binary topological product. If 𝑅 and 𝑆 are known to be topologies, see txtop 15284. (Contributed by Jim Kingdon, 3-Aug-2023.)

Assertion

Ref	Expression
txvalex	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)

Proof of Theorem txvalex

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2833	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
3		elex 2833	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
5		mpoexga 6438	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
6		rnexg 5042	. . . 4 ⊢ ((𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
7		tgvalex 13594	. . . 4 ⊢ (ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ∈ V → (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))) ∈ V)
8	5, 6, 7	3syl 17	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))) ∈ V)
9		mpoeq12 6138	. . . . . 6 ⊢ ((𝑤 = 𝑅 ∧ 𝑧 = 𝑆) → (𝑥 ∈ 𝑤, 𝑦 ∈ 𝑧 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
10	9	rneqd 5006	. . . . 5 ⊢ ((𝑤 = 𝑅 ∧ 𝑧 = 𝑆) → ran (𝑥 ∈ 𝑤, 𝑦 ∈ 𝑧 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
11	10	fveq2d 5694	. . . 4 ⊢ ((𝑤 = 𝑅 ∧ 𝑧 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑤, 𝑦 ∈ 𝑧 ↦ (𝑥 × 𝑦))) = (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))))
12		df-tx 15277	. . . 4 ⊢ ×t = (𝑤 ∈ V, 𝑧 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑤, 𝑦 ∈ 𝑧 ↦ (𝑥 × 𝑦))))
13	11, 12	ovmpoga 6208	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))) ∈ V) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))))
14	2, 4, 8, 13	syl3anc 1278	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))))
15	14, 8	eqeltrd 2315	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   × cxp 4767  ran crn 4770  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  topGenctg 13585   ×_t ctx 15276

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-topgen 13591  df-tx 15277

This theorem is referenced by:  txbasval  15291