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Mirrors > Home > MPE Home > Th. List > Mathboxes > idinxpssinxp2 | Structured version Visualization version GIF version |
Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
idinxpssinxp2 | ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idinxpresid 5908 | . . . 4 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴) | |
2 | 1 | sseq1i 3993 | . . 3 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴))) |
3 | idrefALT 5966 | . . 3 ⊢ (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥) | |
4 | brinxp2 5622 | . . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥)) | |
5 | pm4.24 566 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | 5 | anbi1i 625 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥)) |
7 | 4, 6 | bitr4i 280 | . . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥)) |
8 | 7 | ralbii 3163 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥)) |
9 | 2, 3, 8 | 3bitri 299 | . 2 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥)) |
10 | ralanid 3166 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | |
11 | 9, 10 | bitri 277 | 1 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2108 ∀wral 3136 ∩ cin 3933 ⊆ wss 3934 class class class wbr 5057 I cid 5452 × cxp 5546 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: idinxpssinxp3 35568 idinxpssinxp4 35569 |
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