Theorem txval 21177

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 3185	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		elex 3185	. 2 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
3		mpt2eq12 6613	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
4	3	rneqd 5274	. . . . 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	syl6eqr 2662	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
7	6	fveq2d 6107	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8		df-tx 21175	. . 3 ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
9		fvex 6113	. . 3 ⊢ (topGen‘𝐵) ∈ V
10	7, 8, 9	ovmpt2a 6689	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
11	1, 2, 10	syl2an 493	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   × cxp 5036  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  topGenctg 15921   ×_t ctx 21173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-tx 21175

This theorem is referenced by:  eltx  21181  txtop  21182  txtopon  21204  txopn  21215  txss12  21218  txbasval  21219  txcnp  21233  txcnmpt  21237  txrest  21244  txlm  21261  tx2ndc  21264  txflf  21620  mbfimaopnlem  23228