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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 6861 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋) | |
3 | f1ofn 6850 | . . . 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 6861 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌) | |
7 | f1of 6849 | . . . 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 6129 | . . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷) | |
11 | 10 | relbrcnv 6128 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵) |
12 | cnvco 5899 | . . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶) | |
13 | 12 | breqi 5154 | . . . . 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 44020 | . 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴)) |
17 | f1orel 6852 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶) | |
18 | relbrcnvg 6126 | . . . 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 6852 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷) | |
21 | relbrcnvg 6126 | . . . 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 5148 ◡ccnv 5688 ∘ ccom 5693 Rel wrel 5694 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: (None) |
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