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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 3070 | . . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) | |
| 2 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | elima 6082 | . . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥) | 
| 4 | vex 3483 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 5 | 4, 2 | brco 5880 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | 
| 6 | 5 | rexbii 3093 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | 
| 7 | rexcom4 3287 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
| 8 | r19.41v 3188 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
| 9 | 8 | exbii 1847 | . . . . 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 6082 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥) | 
| 12 | vex 3483 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 13 | 12 | elima 6082 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦) | 
| 14 | 13 | anbi1i 624 | . . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | 
| 15 | 14 | exbii 1847 | . . . 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 2733 | 1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 class class class wbr 5142 “ cima 5687 ∘ ccom 5688 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: fvco2 7005 suppco 8232 fipreima 9399 fsuppcolem 9442 psgnunilem1 19512 gsumzf1o 19931 dprdf1o 20053 frlmup3 21821 f1lindf 21843 lindfmm 21848 cnco 23275 cnpco 23276 ptrescn 23648 xkoco1cn 23666 xkoco2cn 23667 xkococnlem 23668 qtopcn 23723 fmco 23970 uniioombllem3 25621 cncombf 25694 deg1val 26136 ofpreima 32676 mbfmco 34267 eulerpartlemmf 34378 erdsze2lem2 35210 cvmliftmolem1 35287 cvmlift2lem9a 35309 cvmlift2lem9 35317 mclsppslem 35589 bj-imdirco 37192 poimirlem15 37643 poimirlem16 37644 poimirlem19 37647 cnambfre 37676 ftc1anclem3 37703 aks6d1c6lem4 42175 aks6d1c6lem5 42179 trclimalb2 43744 brtrclfv2 43745 frege97d 43770 frege109d 43775 frege131d 43782 extoimad 44182 imo72b2lem0 44183 imo72b2lem2 44185 imo72b2lem1 44187 imo72b2 44190 limccog 45640 smfco 46822 afv2co2 47274 grimco 47885 | 
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