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Mirrors > Home > MPE Home > Th. List > ptunimpt | Structured version Visualization version GIF version |
Description: Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
ptunimpt.j | ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾)) |
Ref | Expression |
---|---|
ptunimpt | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐾) = (𝑥 ∈ 𝐴 ↦ 𝐾) | |
2 | 1 | fvmpt2 6807 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) = 𝐾) |
3 | 2 | eqcomd 2742 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → 𝐾 = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
4 | 3 | unieqd 4819 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐾 ∈ Top) → ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
5 | 4 | ralimiaa 3072 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
6 | 5 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → ∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
7 | ixpeq2 8570 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
9 | nffvmpt1 6706 | . . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) | |
10 | 9 | nfuni 4812 | . . . 4 ⊢ Ⅎ𝑥∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) |
11 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑦∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) | |
12 | fveq2 6695 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) | |
13 | 12 | unieqd 4819 | . . . 4 ⊢ (𝑦 = 𝑥 → ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥)) |
14 | 10, 11, 13 | cbvixp 8573 | . . 3 ⊢ X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = X𝑥 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑥) |
15 | 8, 14 | eqtr4di 2789 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦)) |
16 | 1 | fmpt 6905 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐾 ∈ Top ↔ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) |
17 | ptunimpt.j | . . . 4 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾)) | |
18 | 17 | ptuni 22445 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐾):𝐴⟶Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽) |
19 | 16, 18 | sylan2b 597 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑦 ∈ 𝐴 ∪ ((𝑥 ∈ 𝐴 ↦ 𝐾)‘𝑦) = ∪ 𝐽) |
20 | 15, 19 | eqtrd 2771 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∪ cuni 4805 ↦ cmpt 5120 ⟶wf 6354 ‘cfv 6358 Xcixp 8556 ∏tcpt 16897 Topctop 21744 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-1o 8180 df-er 8369 df-ixp 8557 df-en 8605 df-fin 8608 df-fi 9005 df-topgen 16902 df-pt 16903 df-top 21745 df-bases 21797 |
This theorem is referenced by: pttopon 22447 kelac1 40532 |
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