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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 7552 | . 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 7436 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} | |
3 | 2 | reseq1i 5996 | . 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) |
4 | df-mpo 7436 | . 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)} | |
5 | 1, 3, 4 | 3eqtr4g 2800 | 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 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 × cxp 5687 ↾ cres 5691 {coprab 7432 ∈ cmpo 7433 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 df-res 5701 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: elimampo 7570 ofmres 8008 cantnfval2 9707 submefmnd 18921 pgrpsubgsymg 19442 sylow3lem5 19664 rhmsubclem1 20702 phssip 21694 mamures 22417 mdetrsca2 22626 mdetrlin2 22629 mdetunilem5 22638 smadiadetglem1 22693 smadiadetglem2 22694 pmatcollpw3lem 22805 txss12 23629 txbasval 23630 cnmpt2res 23701 fmucndlem 24316 cnmpopc 24969 oprpiece1res1 24996 oprpiece1res2 24997 cxpcn3 26806 ressplusf 32933 submatres 33767 cvmlift2lem6 35293 cvmlift2lem12 35299 icorempo 37334 elicores 45486 volicorescl 46509 rngchomrnghmresALTV 48123 rhmsubcALTVlem1 48125 |
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