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Mirrors > Home > MPE Home > Th. List > resmpo | Structured version Visualization version GIF version |
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.) |
Ref | Expression |
---|---|
resmpo | ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resoprab2 7268 | . 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)}) | |
2 | df-mpo 7158 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} | |
3 | 2 | reseq1i 5846 | . 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) |
4 | df-mpo 7158 | . 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)} | |
5 | 1, 3, 4 | 3eqtr4g 2880 | 1 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3933 × cxp 5550 ↾ cres 5554 {coprab 7154 ∈ cmpo 7155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-opab 5126 df-xp 5558 df-rel 5559 df-res 5564 df-oprab 7157 df-mpo 7158 |
This theorem is referenced by: ofmres 7682 cantnfval2 9129 submefmnd 18056 pgrpsubgsymg 18533 sylow3lem5 18752 phssip 20798 mamures 20997 mdetrsca2 21209 mdetrlin2 21212 mdetunilem5 21221 smadiadetglem1 21276 smadiadetglem2 21277 pmatcollpw3lem 21387 txss12 22209 txbasval 22210 cnmpt2res 22281 fmucndlem 22896 cnmpopc 23528 oprpiece1res1 23551 oprpiece1res2 23552 cxpcn3 25327 ressplusf 30637 submatres 31095 cvmlift2lem6 32579 cvmlift2lem12 32585 icorempo 34659 elicores 41883 volicorescl 42909 rngchomrnghmresALTV 44341 rhmsubclem1 44431 rhmsubcALTVlem1 44449 |
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