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Mirrors > Home > MPE Home > Th. List > tskmval | Structured version Visualization version GIF version |
Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
tskmval | ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3515 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | grothtsk 10260 | . . . . 5 ⊢ ∪ Tarski = V | |
3 | 1, 2 | eleqtrrdi 2927 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ Tarski) |
4 | eluni2 4845 | . . . 4 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) | |
5 | 3, 4 | sylib 220 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥) |
6 | intexrab 5246 | . . 3 ⊢ (∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ↔ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) | |
7 | 5, 6 | sylib 220 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) |
8 | eleq1 2903 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
9 | 8 | rabbidv 3483 | . . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
10 | 9 | inteqd 4884 | . . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥} = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
11 | df-tskm 10263 | . . 3 ⊢ tarskiMap = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ Tarski ∣ 𝑦 ∈ 𝑥}) | |
12 | 10, 11 | fvmptg 6769 | . 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ V) → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
13 | 1, 7, 12 | syl2anc 586 | 1 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 {crab 3145 Vcvv 3497 ∪ cuni 4841 ∩ cint 4879 ‘cfv 6358 Tarskictsk 10173 tarskiMapctskm 10262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-groth 10248 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-tsk 10174 df-tskm 10263 |
This theorem is referenced by: tskmid 10265 tskmcl 10266 sstskm 10267 eltskm 10268 |
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