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Mirrors > Home > MPE Home > Th. List > fin1a2lem3 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9908. (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 7172 | . 2 ⊢ (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴)) | |
2 | fin1a2lem.b | . . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) | |
3 | oveq2 7172 | . . . 4 ⊢ (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎)) | |
4 | 3 | cbvmptv 5130 | . . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎)) |
5 | 2, 4 | eqtri 2761 | . 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎)) |
6 | ovex 7197 | . 2 ⊢ (2o ·o 𝐴) ∈ V | |
7 | 1, 5, 6 | fvmpt 6769 | 1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 ωcom 7593 2oc2o 8118 ·o comu 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fv 6341 df-ov 7167 |
This theorem is referenced by: fin1a2lem4 9896 fin1a2lem5 9897 fin1a2lem6 9898 |
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