Theorem tposfn2 6375

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 6369	. . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹)
2	1	a1i 9	. . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹))
3		dmtpos 6365	. . . . . 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 4775	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
6		cnveq 4870	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴)
7	6	eqeq2d 2219	. . . . 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 5293	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
12		df-fn 5293	. 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 1373  ^◡ccnv 4692  dom cdm 4693  Rel wrel 4698  Fun wfun 5284   Fn wfn 5285  tpos ctpos 6353

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-tpos 6354

This theorem is referenced by:  tposfo2  6376  tpos0  6383