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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1ocan1fv | Structured version Visualization version GIF version |
Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
f1ocan1fv | ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of 6827 | . . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶𝐵) | |
2 | 1 | 3ad2ant2 1131 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝐺:𝐴⟶𝐵) |
3 | f1ocnv 6839 | . . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴) | |
4 | f1of 6827 | . . . . . 6 ⊢ (◡𝐺:𝐵–1-1-onto→𝐴 → ◡𝐺:𝐵⟶𝐴) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵⟶𝐴) |
6 | 5 | 3ad2ant2 1131 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ◡𝐺:𝐵⟶𝐴) |
7 | simp3 1135 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
8 | 6, 7 | ffvelcdmd 7081 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (◡𝐺‘𝑋) ∈ 𝐴) |
9 | fvco3 6984 | . . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (◡𝐺‘𝑋) ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋)))) | |
10 | 2, 8, 9 | syl2anc 583 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋)))) |
11 | f1ocnvfv2 7271 | . . . 4 ⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋) | |
12 | 11 | 3adant1 1127 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋) |
13 | 12 | fveq2d 6889 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘(◡𝐺‘𝑋))) = (𝐹‘𝑋)) |
14 | 10, 13 | eqtrd 2766 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ◡ccnv 5668 ∘ ccom 5673 Fun wfun 6531 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 |
This theorem is referenced by: f1ocan2fv 37108 |
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