Theorem tposfo2 6116

Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposfo2	⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))

Proof of Theorem tposfo2

Step	Hyp	Ref	Expression
1		tposfn2 6115	. . . 4 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))
2	1	adantrd 275	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn ◡𝐴))
3		fndm 5178	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	releqd 4581	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
5	4	biimparc 295	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → Rel dom 𝐹)
6		rntpos 6106	. . . . . . 7 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
7	5, 6	syl 14	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
8	7	eqeq1d 2121	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
9	8	biimprd 157	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
10	9	expimpd 358	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
11	2, 10	jcad 303	. 2 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12		df-fo 5085	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13		df-fo 5085	. 2 ⊢ (tpos 𝐹:◡𝐴–onto→𝐵 ↔ (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵))
14	11, 12, 13	3imtr4g 204	1 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1312  ^◡ccnv 4496  dom cdm 4497  ran crn 4498  Rel wrel 4502   Fn wfn 5074  –onto→wfo 5077  tpos ctpos 6093

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-fo 5085  df-fv 5087  df-tpos 6094

This theorem is referenced by:  tposf2  6117  tposf1o2  6119  tposfo  6120