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Mirrors > Home > MPE Home > Th. List > Mathboxes > intimass2 | Structured version Visualization version GIF version |
Description: The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) |
Ref | Expression |
---|---|
intimass2 | ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intimass 41908 | . 2 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)} | |
2 | intima0 41902 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)} | |
3 | 1, 2 | sseqtrri 3981 | 1 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 {cab 2713 ∃wrex 3073 ⊆ wss 3910 ∩ cint 4907 ∩ ciin 4955 “ cima 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iin 4957 df-br 5106 df-opab 5168 df-xp 5639 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 |
This theorem is referenced by: (None) |
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