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Mirrors > Home > MPE Home > Th. List > rnin | Structured version Visualization version GIF version |
Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
rnin | ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvin 5998 | . . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) | |
2 | 1 | dmeqi 5768 | . . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵) |
3 | dmin 5775 | . . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) | |
4 | 2, 3 | eqsstri 4001 | . 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) |
5 | df-rn 5561 | . 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵) | |
6 | df-rn 5561 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5561 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | ineq12i 4187 | . 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵) |
9 | 4, 5, 8 | 3sstr4i 4010 | 1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3935 ⊆ wss 3936 ◡ccnv 5549 dom cdm 5550 ran crn 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 |
This theorem is referenced by: inimass 6007 restutop 22840 |
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