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Mirrors > Home > MPE Home > Th. List > mpoeq12 | Structured version Visualization version GIF version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpoeq12 | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 ⊢ 𝐸 = 𝐸 | |
2 | 1 | rgenw 3073 | . . . 4 ⊢ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸 |
3 | 2 | jctr 528 | . . 3 ⊢ (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) |
4 | 3 | ralrimivw 3106 | . 2 ⊢ (𝐵 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) |
5 | mpoeq123 7283 | . 2 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) | |
6 | 4, 5 | sylan2 596 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∀wral 3061 ∈ cmpo 7215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-oprab 7217 df-mpo 7218 |
This theorem is referenced by: dffi3 9047 cantnfres 9292 xpsval 17075 monpropd 17242 grpsubpropd2 18469 lsmvalx 19028 lsmpropd 19067 psrplusgpropd 21157 d1mat2pmat 21636 txval 22461 cnmptk1p 22582 cnmptk2 22583 xpstopnlem1 22706 rrxmval 24302 madjusmdetlem1 31491 pstmval 31559 qqhval2 31644 funcrngcsetc 45229 funcringcsetc 45266 lmod1zr 45507 |
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