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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 6053 | . . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺) | |
2 | dfrel2 5502 | . . . . . 6 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺) | |
3 | 1, 2 | sylib 207 | . . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡◡𝐺 = 𝐺) |
4 | 3 | 3ad2ant2 1076 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ◡◡𝐺 = 𝐺) |
5 | 4 | fveq1d 6105 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (◡◡𝐺‘𝑋) = (𝐺‘𝑋)) |
6 | 5 | fveq2d 6107 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋))) |
7 | f1ocnv 6062 | . . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴) | |
8 | f1ocan1fv 32691 | . . 3 ⊢ ((Fun 𝐹 ∧ ◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) | |
9 | 7, 8 | syl3an2 1352 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(◡◡𝐺‘𝑋)) = (𝐹‘𝑋)) |
10 | 6, 9 | eqtr3d 2646 | 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 1031 = wceq 1475 ∈ wcel 1977 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798 –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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