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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 3400 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | elex 3400 | . 2 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V) | |
3 | mpt2eq12 6949 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))) | |
4 | 3 | rneqd 5556 | . . . . 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 2851 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = 𝐵) |
7 | 6 | fveq2d 6415 | . . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵)) |
8 | df-tx 21694 | . . 3 ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))) | |
9 | fvex 6424 | . . 3 ⊢ (topGen‘𝐵) ∈ V | |
10 | 7, 8, 9 | ovmpt2a 7025 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
11 | 1, 2, 10 | syl2an 590 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 × cxp 5310 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 topGenctg 16413 ×t ctx 21692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-tx 21694 |
This theorem is referenced by: eltx 21700 txtop 21701 txtopon 21723 txopn 21734 txss12 21737 txbasval 21738 txcnp 21752 txcnmpt 21756 txrest 21763 txlm 21780 tx2ndc 21783 txflf 22138 mbfimaopnlem 23763 |
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