Theorem txval 23593

Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Hypothesis

Ref	Expression
txval.1	⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))

Assertion

Ref	Expression
txval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem txval

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3509	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		elex 3509	. 2 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
3		mpoeq12 7523	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
4	3	rneqd 5963	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
5		txval.1	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
6	4, 5	eqtr4di 2798	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
7	6	fveq2d 6924	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8		df-tx 23591	. . 3 ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
9		fvex 6933	. . 3 ⊢ (topGen‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7605	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
11	1, 2, 10	syl2an 595	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   × cxp 5698  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  topGenctg 17497   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-tx 23591

This theorem is referenced by:  eltx  23597  txtop  23598  txtopon  23620  txopn  23631  txss12  23634  txbasval  23635  txcnp  23649  txcnmpt  23653  txrest  23660  txlm  23677  tx2ndc  23680  txflf  24035  mbfimaopnlem  25709