tposfn2 - Metamath Proof Explorer

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Theorem tposfn2 7419

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 7413	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 7409	. . . . . 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 5235	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5328	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ^◡dom 𝐹 = ^◡𝐴)
7	6	eqeq2d 2661	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ^◡dom 𝐹 ↔ dom tpos 𝐹 = ^◡𝐴))
8	4, 5, 7	3imtr3d 282	. . . 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 585	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴)))
11		df-fn 5929	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 5929	. 2 ⊢ (tpos 𝐹 Fn ^◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ^◡𝐴))
13	10, 11, 12	3imtr4g 285	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ^◡𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ^◡ccnv 5142 dom cdm 5143 Rel wrel 5148 Fun wfun 5920 Fn wfn 5921 tpos ctpos 7396

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 df-tpos 7397

This theorem is referenced by: tposfo2 7420 tpos0 7427