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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.27dlem1 | Structured version Visualization version GIF version |
Description: Lemma for rmydioph 41743. 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 2788 | 1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ↾ cres 5678 ‘cfv 6543 (class class class)co 7408 1c1 11110 ...cfz 13483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-res 5688 df-iota 6495 df-fv 6551 |
This theorem is referenced by: rmydioph 41743 rmxdioph 41745 expdiophlem2 41751 |
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