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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapco2 | Structured version Visualization version GIF version |
Description: Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
Ref | Expression |
---|---|
mapco2.3 | ⊢ 𝐸 ∈ V |
Ref | Expression |
---|---|
mapco2 | ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapco2.3 | . 2 ⊢ 𝐸 ∈ V | |
2 | mapco2g 42004 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) | |
3 | 1, 2 | mp3an1 1444 | 1 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 Vcvv 3466 ∘ ccom 5671 ⟶wf 6530 (class class class)co 7402 ↑m cmap 8817 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 |
This theorem is referenced by: diophren 42103 rabrenfdioph 42104 |
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