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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 6627 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋) | |
3 | f1ofn 6616 | . . . 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 6627 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌) | |
7 | f1of 6615 | . . . 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 6097 | . . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷) | |
11 | 10 | relbrcnv 5970 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵) |
12 | cnvco 5756 | . . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶) | |
13 | 12 | breqi 5072 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
14 | 11, 13 | bitr3i 279 | . . . 4 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
15 | 9, 14 | sylib 220 | . . 3 ⊢ (𝜑 → 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
16 | 4, 8, 15 | brcoffn 40400 | . 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴)) |
17 | f1orel 6618 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶) | |
18 | relbrcnvg 5968 | . . . 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 6618 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷) | |
21 | relbrcnvg 5968 | . . . 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 234 | 1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 class class class wbr 5066 ◡ccnv 5554 ∘ ccom 5559 Rel wrel 5560 Fn wfn 6350 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 |
This theorem is referenced by: (None) |
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