![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > f1ocan2fv | Structured version Visualization version GIF version |
Description: Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
f1ocan2fv | ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1orel 6791 | . . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺) | |
2 | dfrel2 6145 | . . . . . 6 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡◡𝐺 = 𝐺) |
4 | 3 | 3ad2ant2 1135 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ◡◡𝐺 = 𝐺) |
5 | 4 | fveq1d 6848 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (◡◡𝐺‘𝑋) = (𝐺‘𝑋)) |
6 | 5 | fveq2d 6850 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋))) |
7 | f1ocnv 6800 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴) | |
8 | f1ocan1fv 36235 | . . 3 ⊢ ((Fun 𝐹 ∧ ◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) | |
9 | 7, 8 | syl3an2 1165 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) |
10 | 6, 9 | eqtr3d 2775 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ◡ccnv 5636 ∘ ccom 5641 Rel wrel 5642 Fun wfun 6494 –1-1-onto→wf1o 6499 ‘cfv 6500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |