![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2902 | . . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) | |
2 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | elima 5390 | . . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥) |
4 | rexcom4 3198 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
5 | r19.41v 3070 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
6 | 5 | exbii 1764 | . . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
7 | 4, 6 | bitri 263 | . . . 4 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
8 | 2 | elima 5390 | . . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥) |
9 | vex 3176 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
10 | 9, 2 | brco 5214 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
11 | 10 | rexbii 3023 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
12 | 8, 11 | bitri 263 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
13 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | elima 5390 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦) |
15 | 14 | anbi1i 727 | . . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
16 | 15 | exbii 1764 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
17 | 7, 12, 16 | 3bitr4i 291 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) |
18 | 1, 3, 17 | 3bitr4ri 292 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶))) |
19 | 18 | eqriv 2607 | 1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 “ cima 5041 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: fvco2 6183 supp0cosupp0 7221 imacosupp 7222 fipreima 8155 fsuppcolem 8189 psgnunilem1 17736 gsumzf1o 18136 dprdf1o 18254 frlmup3 19958 f1lindf 19980 lindfmm 19985 cnco 20880 cnpco 20881 ptrescn 21252 xkoco1cn 21270 xkoco2cn 21271 xkococnlem 21272 qtopcn 21327 fmco 21575 uniioombllem3 23159 cncombf 23231 deg1val 23660 ofpreima 28848 mbfmco 29653 eulerpartlemmf 29764 erdsze2lem2 30440 cvmliftmolem1 30517 cvmlift2lem9a 30539 cvmlift2lem9 30547 mclsppslem 30734 poimirlem15 32594 poimirlem16 32595 poimirlem19 32598 cnambfre 32628 ftc1anclem3 32657 trclimalb2 37037 brtrclfv2 37038 frege97d 37063 frege109d 37068 frege131d 37075 extoimad 37486 imo72b2lem0 37487 imo72b2lem2 37489 imo72b2lem1 37493 imo72b2 37497 limccog 38687 smfco 39687 |
Copyright terms: Public domain | W3C validator |