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Mirrors > Home > MPE Home > Th. List > Mathboxes > relintab | Structured version Visualization version GIF version |
Description: Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
relintab | ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv 6042 | . . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) | |
2 | incom 4175 | . . 3 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | eqtri 2841 | . 2 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) |
4 | dfrel2 6039 | . . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} ↔ ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) | |
5 | 4 | biimpi 217 | . 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ◡◡∩ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) |
6 | relintabex 39819 | . . . 4 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) | |
7 | 6 | xpinintabd 39818 | . . 3 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)}) |
8 | incom 4175 | . . . . . . . . 9 ⊢ ((V × V) ∩ 𝑥) = (𝑥 ∩ (V × V)) | |
9 | cnvcnv 6042 | . . . . . . . . 9 ⊢ ◡◡𝑥 = (𝑥 ∩ (V × V)) | |
10 | 8, 9 | eqtr4i 2844 | . . . . . . . 8 ⊢ ((V × V) ∩ 𝑥) = ◡◡𝑥 |
11 | 10 | eqeq2i 2831 | . . . . . . 7 ⊢ (𝑤 = ((V × V) ∩ 𝑥) ↔ 𝑤 = ◡◡𝑥) |
12 | 11 | anbi1i 623 | . . . . . 6 ⊢ ((𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ (𝑤 = ◡◡𝑥 ∧ 𝜑)) |
13 | 12 | exbii 1839 | . . . . 5 ⊢ (∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑) ↔ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)) |
14 | 13 | rabbii 3471 | . . . 4 ⊢ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)} |
15 | 14 | inteqi 4871 | . . 3 ⊢ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ((V × V) ∩ 𝑥) ∧ 𝜑)} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)} |
16 | 7, 15 | syl6eq 2869 | . 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}) |
17 | 3, 5, 16 | 3eqtr3a 2877 | 1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 {cab 2796 {crab 3139 Vcvv 3492 ∩ cin 3932 𝒫 cpw 4535 ∩ cint 4867 × cxp 5546 ◡ccnv 5547 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-int 4868 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 |
This theorem is referenced by: (None) |
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