Theorem tposfn2 7639

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 7633	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 11	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 7629	. . . . . 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 5436	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 5528	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴)
7	6	eqeq2d 2835	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡𝐴))
8	4, 5, 7	3imtr3d 285	. . . 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 604	. 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴)))
11		df-fn 6126	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 6126	. 2 ⊢ (tpos 𝐹 Fn ◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴))
13	10, 11, 12	3imtr4g 288	1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658  ^◡ccnv 5341  dom cdm 5342  Rel wrel 5347  Fun wfun 6117   Fn wfn 6118  tpos ctpos 7616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-tpos 7617

This theorem is referenced by:  tposfo2  7640  tpos0  7647