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| Mirrors > Home > ILE Home > Th. List > imaco | 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 2517 | . . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) | |
| 2 | vex 2806 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | elima 5087 | . . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥) |
| 4 | rexcom4 2827 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
| 5 | r19.41v 2690 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) | |
| 6 | 5 | exbii 1654 | . . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 7 | 4, 6 | bitri 184 | . . . 4 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 8 | 2 | elima 5087 | . . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥) |
| 9 | vex 2806 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 10 | 9, 2 | brco 4907 | . . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 11 | 10 | rexbii 2540 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 12 | 8, 11 | bitri 184 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 13 | vex 2806 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 14 | 13 | elima 5087 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦) |
| 15 | 14 | anbi1i 458 | . . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 16 | 15 | exbii 1654 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥)) |
| 17 | 7, 12, 16 | 3bitr4i 212 | . . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥)) |
| 18 | 1, 3, 17 | 3bitr4ri 213 | . 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶))) |
| 19 | 18 | eqriv 2228 | 1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶)) |
| Colors of variables: wff set class |
| Syntax hints: ∧ wa 104 = wceq 1398 ∃wex 1541 ∈ wcel 2202 ∃wrex 2512 class class class wbr 4093 “ cima 4734 ∘ ccom 4735 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 |
| This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-br 4094 df-opab 4156 df-xp 4737 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 |
| This theorem is referenced by: fvco2 5724 suppcofn 6444 cnco 15015 cnptopco 15016 |
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