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Mirrors > Home > MPE Home > Th. List > imaco | Structured version Visualization version GIF version |
Description: Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.) |
Ref | Expression |
---|---|
imaco | ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3069 | . . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) | |
2 | vex 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elima 6085 | . . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥) |
4 | vex 3482 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
5 | 4, 2 | brco 5884 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
6 | 5 | rexbii 3092 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
7 | rexcom4 3286 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
8 | r19.41v 3187 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
9 | 8 | exbii 1845 | . . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
10 | 6, 7, 9 | 3bitri 297 | . . . 4 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
11 | 2 | elima 6085 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥) |
12 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
13 | 12 | elima 6085 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦) |
14 | 13 | anbi1i 624 | . . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
15 | 14 | exbii 1845 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
16 | 10, 11, 15 | 3bitr4i 303 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) |
17 | 1, 3, 16 | 3bitr4ri 304 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶))) |
18 | 17 | eqriv 2732 | 1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 “ cima 5692 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: fvco2 7006 suppco 8230 fipreima 9396 fsuppcolem 9439 psgnunilem1 19526 gsumzf1o 19945 dprdf1o 20067 frlmup3 21838 f1lindf 21860 lindfmm 21865 cnco 23290 cnpco 23291 ptrescn 23663 xkoco1cn 23681 xkoco2cn 23682 xkococnlem 23683 qtopcn 23738 fmco 23985 uniioombllem3 25634 cncombf 25707 deg1val 26150 ofpreima 32682 mbfmco 34246 eulerpartlemmf 34357 erdsze2lem2 35189 cvmliftmolem1 35266 cvmlift2lem9a 35288 cvmlift2lem9 35296 mclsppslem 35568 bj-imdirco 37173 poimirlem15 37622 poimirlem16 37623 poimirlem19 37626 cnambfre 37655 ftc1anclem3 37682 aks6d1c6lem4 42155 aks6d1c6lem5 42159 trclimalb2 43716 brtrclfv2 43717 frege97d 43742 frege109d 43747 frege131d 43754 extoimad 44154 imo72b2lem0 44155 imo72b2lem2 44157 imo72b2lem1 44159 imo72b2 44162 limccog 45576 smfco 46758 afv2co2 47207 grimco 47818 |
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