Theorem txval 23551

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 3454	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		elex 3454	. 2 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
3		mpoeq12 7433	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)))
4	3	rneqd 5887	. . . . 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 2794	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
7	6	fveq2d 6835	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8		df-tx 23549	. . 3 ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
9		fvex 6844	. . 3 ⊢ (topGen‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7515	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
11	1, 2, 10	syl2an 603	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433   × cxp 5619  ran crn 5622  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362  topGenctg 17395   ×_t ctx 23547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-tx 23549

This theorem is referenced by:  eltx  23555  txtop  23556  txtopon  23578  txopn  23589  txss12  23592  txbasval  23593  txcnp  23607  txcnmpt  23611  txrest  23618  txlm  23635  tx2ndc  23638  txflf  23993  mbfimaopnlem  25644