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| Mirrors > Home > MPE Home > Th. List > fin1a2lem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
| Ref | Expression |
|---|---|
| fin1a2lem.b | ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) |
| Ref | Expression |
|---|---|
| fin1a2lem3 | ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 7439 | . 2 ⊢ (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴)) | |
| 2 | fin1a2lem.b | . . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) | |
| 3 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎)) | |
| 4 | 3 | cbvmptv 5255 | . . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎)) |
| 5 | 2, 4 | eqtri 2765 | . 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎)) |
| 6 | ovex 7464 | . 2 ⊢ (2o ·o 𝐴) ∈ V | |
| 7 | 1, 5, 6 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ωcom 7887 2oc2o 8500 ·o comu 8504 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: fin1a2lem4 10443 fin1a2lem5 10444 fin1a2lem6 10445 |
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