f1ocan2fv - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > f1ocan2fv		Structured version Visualization version GIF version

Theorem f1ocan2fv 38661

Description: Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
f1ocan2fv	⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ^◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))

**Proof of Theorem f1ocan2fv**
Step	Hyp	Ref	Expression
1		f1orel 6827	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺)
2		dfrel2 6181	. . . . . 6 ⊢ (Rel 𝐺 ↔ ^◡^◡𝐺 = 𝐺)
3	1, 2	sylib 221	. . . . 5 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ^◡^◡𝐺 = 𝐺)
4	3	3ad2ant2 1152	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ^◡^◡𝐺 = 𝐺)
5	4	fveq1d 6887	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → (^◡^◡𝐺‘𝑋) = (𝐺‘𝑋))
6	5	fveq2d 6889	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ^◡𝐺)‘(^◡^◡𝐺‘𝑋)) = ((𝐹 ∘ ^◡𝐺)‘(𝐺‘𝑋)))
7		f1ocnv 6837	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → ^◡𝐺:𝐵–1-1-onto→𝐴)
8		f1ocan1fv 38660	. . 3 ⊢ ((Fun 𝐹 ∧ ^◡𝐺:𝐵–1-1-onto→𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ^◡𝐺)‘(^◡^◡𝐺‘𝑋)) = (𝐹‘𝑋))
9	7, 8	syl3an2 1182	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ^◡𝐺)‘(^◡^◡𝐺‘𝑋)) = (𝐹‘𝑋))
10	6, 9	eqtr3d 2798	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ^◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ^◡ccnv 5650 ∘ ccom 5655 Rel wrel 5656 Fun wfun 6532 –1-1-onto→wf1o 6537 ‘cfv 6538

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546

This theorem is used by: (None)

W3C validator