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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 6617 | . . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺) | |
2 | dfrel2 6045 | . . . . . 6 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺) | |
3 | 1, 2 | sylib 220 | . . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡◡𝐺 = 𝐺) |
4 | 3 | 3ad2ant2 1130 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ◡◡𝐺 = 𝐺) |
5 | 4 | fveq1d 6671 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (◡◡𝐺‘𝑋) = (𝐺‘𝑋)) |
6 | 5 | fveq2d 6673 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋))) |
7 | f1ocnv 6626 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴) | |
8 | f1ocan1fv 35000 | . . 3 ⊢ ((Fun 𝐹 ∧ ◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) | |
9 | 7, 8 | syl3an2 1160 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) |
10 | 6, 9 | eqtr3d 2858 | 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 1083 = wceq 1533 ∈ wcel 2110 ◡ccnv 5553 ∘ ccom 5558 Rel wrel 5559 Fun wfun 6348 –1-1-onto→wf1o 6353 ‘cfv 6354 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 |
This theorem is referenced by: (None) |
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