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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 | ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) |
Ref | Expression |
---|---|
txval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3512 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | elex 3512 | . 2 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V) | |
3 | mpoeq12 7227 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))) | |
4 | 3 | rneqd 5808 | . . . . 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 2874 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = 𝐵) |
7 | 6 | fveq2d 6674 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵)) |
8 | df-tx 22170 | . . 3 ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))) | |
9 | fvex 6683 | . . 3 ⊢ (topGen‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpoa 7305 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
11 | 1, 2, 10 | syl2an 597 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 × cxp 5553 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 topGenctg 16711 ×t ctx 22168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-tx 22170 |
This theorem is referenced by: eltx 22176 txtop 22177 txtopon 22199 txopn 22210 txss12 22213 txbasval 22214 txcnp 22228 txcnmpt 22232 txrest 22239 txlm 22256 tx2ndc 22259 txflf 22614 mbfimaopnlem 24256 |
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