Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > resmpt2 | Structured version Visualization version GIF version |
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.) |
Ref | Expression |
---|---|
resmpt2 | ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resoprab2 6655 | . 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-mpt2 6554 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} | |
3 | 2 | reseq1i 5313 | . 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) |
4 | df-mpt2 6554 | . 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)} | |
5 | 1, 3, 4 | 3eqtr4g 2669 | 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 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 × cxp 5036 ↾ cres 5040 {coprab 6550 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-res 5050 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ofmres 7055 cantnfval2 8449 pgrpsubgsymg 17651 sylow3lem5 17869 phssip 19822 mamures 20015 mdetrsca2 20229 mdetrlin2 20232 mdetunilem5 20241 smadiadetglem1 20296 smadiadetglem2 20297 pmatcollpw3lem 20407 txss12 21218 txbasval 21219 cnmpt2res 21290 fmucndlem 21905 cnmpt2pc 22535 oprpiece1res1 22558 oprpiece1res2 22559 cxpcn3 24289 ressplusf 28981 submatres 29200 cvmlift2lem6 30544 cvmlift2lem12 30550 icorempt2 32375 elicores 38607 volicorescl 39443 rngchomrnghmresALTV 41788 rhmsubclem1 41878 rhmsubcALTVlem1 41897 |
Copyright terms: Public domain | W3C validator |