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Mirrors > Home > MPE Home > Th. List > txval | Structured version Visualization version GIF version |
Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
txval.1 | β’ π΅ = ran (π₯ β π , π¦ β π β¦ (π₯ Γ π¦)) |
Ref | Expression |
---|---|
txval | β’ ((π β π β§ π β π) β (π Γt π) = (topGenβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 β’ (π β π β π β V) | |
2 | elex 3490 | . 2 β’ (π β π β π β V) | |
3 | mpoeq12 7493 | . . . . . 6 β’ ((π = π β§ π = π) β (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) = (π₯ β π , π¦ β π β¦ (π₯ Γ π¦))) | |
4 | 3 | rneqd 5940 | . . . . 5 β’ ((π = π β§ π = π) β ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) = ran (π₯ β π , π¦ β π β¦ (π₯ Γ π¦))) |
5 | txval.1 | . . . . 5 β’ π΅ = ran (π₯ β π , π¦ β π β¦ (π₯ Γ π¦)) | |
6 | 4, 5 | eqtr4di 2786 | . . . 4 β’ ((π = π β§ π = π) β ran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)) = π΅) |
7 | 6 | fveq2d 6901 | . . 3 β’ ((π = π β§ π = π) β (topGenβran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦))) = (topGenβπ΅)) |
8 | df-tx 23465 | . . 3 β’ Γt = (π β V, π β V β¦ (topGenβran (π₯ β π, π¦ β π β¦ (π₯ Γ π¦)))) | |
9 | fvex 6910 | . . 3 β’ (topGenβπ΅) β V | |
10 | 7, 8, 9 | ovmpoa 7576 | . 2 β’ ((π β V β§ π β V) β (π Γt π) = (topGenβπ΅)) |
11 | 1, 2, 10 | syl2an 595 | 1 β’ ((π β π β§ π β π) β (π Γt π) = (topGenβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 Γ cxp 5676 ran crn 5679 βcfv 6548 (class class class)co 7420 β cmpo 7422 topGenctg 17418 Γt 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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