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Mirrors > Home > MPE Home > Th. List > inteq | Structured version Visualization version GIF version |
Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.) |
Ref | Expression |
---|---|
inteq | ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 3138 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)) | |
2 | 1 | abbidv 2741 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦}) |
3 | dfint2 4477 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
4 | dfint2 4477 | . 2 ⊢ ∩ 𝐵 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦} | |
5 | 2, 3, 4 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 {cab 2608 ∀wral 2912 ∩ cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-int 4476 |
This theorem is referenced by: inteqi 4479 inteqd 4480 unissint 4501 uniintsn 4514 rint0 4517 intex 4820 intnex 4821 elreldm 5350 elxp5 7111 1stval2 7185 oev2 7603 fundmen 8030 xpsnen 8044 fiint 8237 elfir 8321 inelfi 8324 fiin 8328 cardmin2 8824 isfin2-2 9141 incexclem 14568 mreintcl 16255 ismred2 16263 fiinopn 20706 cmpfii 21212 ptbasfi 21384 fbssint 21642 shintcl 28189 chintcl 28191 inelpisys 30217 rankeq1o 32278 bj-0int 33055 bj-ismoored 33062 bj-snmoore 33068 neificl 33549 heibor1lem 33608 elrfi 37257 elrfirn 37258 |
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