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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.) |
Ref | Expression |
---|---|
imaco | ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3071 | . . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) | |
2 | vex 3451 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elima 6022 | . . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥) |
4 | rexcom4 3270 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
5 | r19.41v 3182 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
6 | 5 | exbii 1851 | . . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
7 | 4, 6 | bitri 275 | . . . 4 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
8 | 2 | elima 6022 | . . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥) |
9 | vex 3451 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
10 | 9, 2 | brco 5830 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
11 | 10 | rexbii 3094 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
12 | 8, 11 | bitri 275 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
13 | vex 3451 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | elima 6022 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦) |
15 | 14 | anbi1i 625 | . . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
16 | 15 | exbii 1851 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
17 | 7, 12, 16 | 3bitr4i 303 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) |
18 | 1, 3, 17 | 3bitr4ri 304 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶))) |
19 | 18 | eqriv 2730 | 1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 class class class wbr 5109 “ cima 5640 ∘ ccom 5641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: fvco2 6942 suppco 8141 fipreima 9308 fsuppcolem 9345 psgnunilem1 19283 gsumzf1o 19697 dprdf1o 19819 frlmup3 21229 f1lindf 21251 lindfmm 21256 cnco 22640 cnpco 22641 ptrescn 23013 xkoco1cn 23031 xkoco2cn 23032 xkococnlem 23033 qtopcn 23088 fmco 23335 uniioombllem3 24972 cncombf 25045 deg1val 25484 ofpreima 31634 mbfmco 32928 eulerpartlemmf 33039 erdsze2lem2 33862 cvmliftmolem1 33939 cvmlift2lem9a 33961 cvmlift2lem9 33969 mclsppslem 34241 bj-imdirco 35711 poimirlem15 36143 poimirlem16 36144 poimirlem19 36147 cnambfre 36176 ftc1anclem3 36203 trclimalb2 42090 brtrclfv2 42091 frege97d 42116 frege109d 42121 frege131d 42128 extoimad 42529 imo72b2lem0 42530 imo72b2lem2 42532 imo72b2lem1 42534 imo72b2 42537 limccog 43951 smfco 45133 afv2co2 45579 isomgrtrlem 46120 |
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