Theorem tposfn2 6321

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 6315	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 9	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 6311	. . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
4	3	a1i 9	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹))
5		releq 4742	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 4837	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴)
7	6	eqeq2d 2205	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡𝐴))
8	4, 5, 7	3imtr3d 202	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = ◡𝐴))
9	8	com12 30	. . 3 ⊢ (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = ◡𝐴))
10	2, 9	anim12d 335	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴)))
11		df-fn 5258	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 5258	. 2 ⊢ (tpos 𝐹 Fn ◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴))
13	10, 11, 12	3imtr4g 205	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ^◡ccnv 4659  dom cdm 4660  Rel wrel 4665  Fun wfun 5249   Fn wfn 5250  tpos ctpos 6299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-tpos 6300

This theorem is referenced by:  tposfo2  6322  tpos0  6329