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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 6101 | . . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) | |
2 | 1 | dmeqi 5864 | . . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵) |
3 | dmin 5871 | . . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) | |
4 | 2, 3 | eqsstri 3982 | . 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) |
5 | df-rn 5648 | . 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵) | |
6 | df-rn 5648 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5648 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | ineq12i 4174 | . 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵) |
9 | 4, 5, 8 | 3sstr4i 3991 | 1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3913 ⊆ wss 3914 ◡ccnv 5636 dom cdm 5637 ran crn 5638 |
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 5260 ax-nul 5267 ax-pr 5388 |
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-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: inimass 6111 restutop 23612 |
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