tposfn2 - Metamath Proof Explorer

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Theorem tposfn2 8188

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 8182	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 8178	. . . . . 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 5724	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5820	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ^◡dom 𝐹 = ^◡𝐴)
7	6	eqeq2d 2745	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ^◡dom 𝐹 ↔ dom tpos 𝐹 = ^◡𝐴))
8	4, 5, 7	3imtr3d 293	. . . 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 6493	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 6493	. 2 ⊢ (tpos 𝐹 Fn ^◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴))
13	10, 11, 12	3imtr4g 296	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ^◡ccnv 5621 dom cdm 5622 Rel wrel 5627 Fun wfun 6484 Fn wfn 6485 tpos ctpos 8165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-fv 6498 df-tpos 8166

This theorem is referenced by: tposfo2 8189 tpos0 8196 tposideq 49075