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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 6062 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋) | |
3 | f1ofn 6051 | . . . 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 6062 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌) | |
7 | f1of 6050 | . . . 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 5550 | . . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷) | |
11 | 10 | relbrcnv 5425 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵) |
12 | cnvco 5230 | . . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶) | |
13 | 12 | breqi 4589 | . . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
14 | 11, 13 | bitr3i 265 | . . . 4 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
15 | 9, 14 | sylib 207 | . . 3 ⊢ (𝜑 → 𝐵(◡𝐷 ∘ ◡𝐶)𝐴) |
16 | 4, 8, 15 | brcoffn 37348 | . 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴)) |
17 | f1orel 6053 | . . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶) | |
18 | relbrcnvg 5423 | . . . 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 6053 | . . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷) | |
21 | relbrcnvg 5423 | . . . 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 743 | . 2 ⊢ (𝜑 → ((𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴) ↔ ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵)))) |
24 | 16, 23 | mpbid 221 | 1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 class class class wbr 4583 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 |
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-8 1979 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-pow 4769 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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: (None) |
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