tposfn2 - Metamath Proof Explorer

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Theorem tposfn2 8232

Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.)

**Assertion**
Ref	Expression
tposfn2	⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

**Proof of Theorem tposfn2**
Step	Hyp	Ref	Expression
1		tposfun 8226	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 8222	. . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ^◡dom 𝐹)
4	3	a1i 11	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = ^◡dom 𝐹))
5		releq 5776	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5873	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ^◡dom 𝐹 = ^◡𝐴)
7	6	eqeq2d 2743	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ^◡dom 𝐹 ↔ dom tpos 𝐹 = ^◡𝐴))
8	4, 5, 7	3imtr3d 292	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = ^◡𝐴))
9	8	com12 32	. . 3 ⊢ (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = ^◡𝐴))
10	2, 9	anim12d 609	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴)))
11		df-fn 6546	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 6546	. 2 ⊢ (tpos 𝐹 Fn ^◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴))
13	10, 11, 12	3imtr4g 295	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ^◡ccnv 5675 dom cdm 5676 Rel wrel 5681 Fun wfun 6537 Fn wfn 6538 tpos ctpos 8209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-tpos 8210

This theorem is referenced by: tposfo2 8233 tpos0 8240