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Mirrors > Home > MPE Home > Th. List > sstskm | Structured version Visualization version GIF version |
Description: Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
sstskm | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tskmval 10864 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | |
2 | df-rab 3419 | . . . . 5 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} | |
3 | 2 | inteqi 4954 | . . . 4 ⊢ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} |
4 | 1, 3 | eqtrdi 2781 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)}) |
5 | 4 | sseq2d 4009 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ 𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)})) |
6 | impexp 449 | . . . 4 ⊢ (((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ (𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) | |
7 | 6 | albii 1813 | . . 3 ⊢ (∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) |
8 | ssintab 4969 | . . 3 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥) → 𝐵 ⊆ 𝑥)) | |
9 | df-ral 3051 | . . 3 ⊢ (∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥) ↔ ∀𝑥(𝑥 ∈ Tarski → (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) | |
10 | 7, 8, 9 | 3bitr4i 302 | . 2 ⊢ (𝐵 ⊆ ∩ {𝑥 ∣ (𝑥 ∈ Tarski ∧ 𝐴 ∈ 𝑥)} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)) |
11 | 5, 10 | bitrdi 286 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 ∈ wcel 2098 {cab 2702 ∀wral 3050 {crab 3418 ⊆ wss 3944 ∩ cint 4950 ‘cfv 6549 Tarskictsk 10773 tarskiMapctskm 10862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-groth 10848 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-tsk 10774 df-tskm 10863 |
This theorem is referenced by: (None) |
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