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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 6608 | . . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴⟶𝐵) | |
2 | 1 | 3ad2ant2 1126 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝐺:𝐴⟶𝐵) |
3 | f1ocnv 6620 | . . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ◡𝐺:𝐵–1-1-onto→𝐴) | |
4 | f1of 6608 | . . . . . 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 1126 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ◡𝐺:𝐵⟶𝐴) |
7 | simp3 1130 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
8 | 6, 7 | ffvelrnd 6844 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (◡𝐺‘𝑋) ∈ 𝐴) |
9 | fvco3 6753 | . . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (◡𝐺‘𝑋) ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋)))) | |
10 | 2, 8, 9 | syl2anc 584 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘(𝐺‘(◡𝐺‘𝑋)))) |
11 | f1ocnvfv2 7025 | . . . 4 ⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋) | |
12 | 11 | 3adant1 1122 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐺‘(◡𝐺‘𝑋)) = 𝑋) |
13 | 12 | fveq2d 6667 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘(◡𝐺‘𝑋))) = (𝐹‘𝑋)) |
14 | 10, 13 | eqtrd 2853 | 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 1079 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 ∘ ccom 5552 Fun wfun 6342 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: f1ocan2fv 34883 |
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