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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.27dlem1 | Structured version Visualization version GIF version |
Description: Lemma for rmydioph 42499. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
Ref | Expression |
---|---|
jm2.27dlem1.1 | ⊢ 𝐴 ∈ (1...𝐵) |
Ref | Expression |
---|---|
jm2.27dlem1 | ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6890 | . 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴)) | |
2 | jm2.27dlem1.1 | . . 3 ⊢ 𝐴 ∈ (1...𝐵) | |
3 | fvres 6910 | . . 3 ⊢ (𝐴 ∈ (1...𝐵) → ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏‘𝐴)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏‘𝐴) |
5 | 1, 4 | eqtrdi 2781 | 1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ↾ cres 5674 ‘cfv 6542 (class class class)co 7415 1c1 11137 ...cfz 13514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-res 5684 df-iota 6494 df-fv 6550 |
This theorem is referenced by: rmydioph 42499 rmxdioph 42501 expdiophlem2 42507 |
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