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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brco2f1o | Structured version Visualization version GIF version | ||
| Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.) |
| Ref | Expression |
|---|---|
| brco2f1o.c | ⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍) |
| brco2f1o.d | ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌) |
| brco2f1o.r | ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵) |
| Ref | Expression |
|---|---|
| brco2f1o | ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brco2f1o.d | . . . 4 ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌) | |
| 2 | f1ocnv 6860 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋) | |
| 3 | f1ofn 6849 | . . . 4 ⊢ (◡𝐷:𝑌–1-1-onto→𝑋 → ◡𝐷 Fn 𝑌) | |
| 4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡𝐷 Fn 𝑌) |
| 5 | brco2f1o.c | . . . 4 ⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍) | |
| 6 | f1ocnv 6860 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌) | |
| 7 | f1of 6848 | . . . 4 ⊢ (◡𝐶:𝑍–1-1-onto→𝑌 → ◡𝐶:𝑍⟶𝑌) | |
| 8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → ◡𝐶:𝑍⟶𝑌) |
| 9 | brco2f1o.r | . . . 4 ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵) | |
| 10 | relco 6126 | . . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷) | |
| 11 | 10 | relbrcnv 6125 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵) |
| 12 | cnvco 5896 | . . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶) | |
| 13 | 12 | breqi 5149 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
| 14 | 11, 13 | bitr3i 277 | . . . 4 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
| 15 | 9, 14 | sylib 218 | . . 3 ⊢ (𝜑 → 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
| 16 | 4, 8, 15 | brcoffn 44043 | . 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴)) |
| 17 | f1orel 6851 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶) | |
| 18 | relbrcnvg 6123 | . . . 4 ⊢ (Rel 𝐶 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵)) | |
| 19 | 5, 17, 18 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵)) |
| 20 | f1orel 6851 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷) | |
| 21 | relbrcnvg 6123 | . . . 4 ⊢ (Rel 𝐷 → ((◡𝐶‘𝐵)◡𝐷𝐴 ↔ 𝐴𝐷(◡𝐶‘𝐵))) | |
| 22 | 1, 20, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → ((◡𝐶‘𝐵)◡𝐷𝐴 ↔ 𝐴𝐷(◡𝐶‘𝐵))) |
| 23 | 19, 22 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴) ↔ ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵)))) |
| 24 | 16, 23 | mpbid 232 | 1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 class class class wbr 5143 ◡ccnv 5684 ∘ ccom 5689 Rel wrel 5690 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: (None) |
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