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Theorem txval 23467
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 𝐡 = ran (π‘₯ ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (π‘₯ Γ— 𝑦))
Assertion
Ref Expression
txval ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝑅 Γ—t 𝑆) = (topGenβ€˜π΅))
Distinct variable groups:   π‘₯,𝑦,𝑅   π‘₯,𝑆,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   π‘Š(π‘₯,𝑦)

Proof of Theorem txval
Dummy variables π‘Ÿ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3490 . 2 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
2 elex 3490 . 2 (𝑆 ∈ π‘Š β†’ 𝑆 ∈ V)
3 mpoeq12 7493 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (π‘₯ ∈ π‘Ÿ, 𝑦 ∈ 𝑠 ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (π‘₯ Γ— 𝑦)))
43rneqd 5940 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ ran (π‘₯ ∈ π‘Ÿ, 𝑦 ∈ 𝑠 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (π‘₯ Γ— 𝑦)))
5 txval.1 . . . . 5 𝐡 = ran (π‘₯ ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (π‘₯ Γ— 𝑦))
64, 5eqtr4di 2786 . . . 4 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ ran (π‘₯ ∈ π‘Ÿ, 𝑦 ∈ 𝑠 ↦ (π‘₯ Γ— 𝑦)) = 𝐡)
76fveq2d 6901 . . 3 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (topGenβ€˜ran (π‘₯ ∈ π‘Ÿ, 𝑦 ∈ 𝑠 ↦ (π‘₯ Γ— 𝑦))) = (topGenβ€˜π΅))
8 df-tx 23465 . . 3 Γ—t = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (topGenβ€˜ran (π‘₯ ∈ π‘Ÿ, 𝑦 ∈ 𝑠 ↦ (π‘₯ Γ— 𝑦))))
9 fvex 6910 . . 3 (topGenβ€˜π΅) ∈ V
107, 8, 9ovmpoa 7576 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) β†’ (𝑅 Γ—t 𝑆) = (topGenβ€˜π΅))
111, 2, 10syl2an 595 1 ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝑅 Γ—t 𝑆) = (topGenβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   Γ— cxp 5676  ran crn 5679  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  topGenctg 17418   Γ—t ctx 23463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-tx 23465
This theorem is referenced by:  eltx  23471  txtop  23472  txtopon  23494  txopn  23505  txss12  23508  txbasval  23509  txcnp  23523  txcnmpt  23527  txrest  23534  txlm  23551  tx2ndc  23554  txflf  23909  mbfimaopnlem  25583
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