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Mirrors > Home > MPE Home > Th. List > Mathboxes > idinxpss | Structured version Visualization version GIF version |
Description: Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.) |
Ref | Expression |
---|---|
idinxpss | ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxpss 37086 | . 2 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦)) | |
2 | ideqg 5846 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
3 | 2 | elv 3481 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | 3 | imbi1i 350 | . . 3 ⊢ ((𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
5 | 4 | 2ralbii 3129 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
6 | 1, 5 | bitri 275 | 1 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wral 3062 Vcvv 3475 ∩ cin 3945 ⊆ wss 3946 class class class wbr 5144 I cid 5569 × cxp 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 |
This theorem is referenced by: ref5 37088 refrelcoss2 37240 dfrefrels3 37290 dfrefrel3 37292 symrefref3 37340 |
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