Theorem tposfo2 6498

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 6497	. . . 4 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))
2	1	adantrd 279	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn ◡𝐴))
3		fndm 5455	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	releqd 4834	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
5	4	biimparc 299	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → Rel dom 𝐹)
6		rntpos 6488	. . . . . . 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 2241	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
9	8	biimprd 158	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
10	9	expimpd 363	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
11	2, 10	jcad 307	. 2 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12		df-fo 5358	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13		df-fo 5358	. 2 ⊢ (tpos 𝐹:◡𝐴–onto→𝐵 ↔ (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵))
14	11, 12, 13	3imtr4g 205	1 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ^◡ccnv 4748  dom cdm 4749  ran crn 4750  Rel wrel 4754   Fn wfn 5347  –onto→wfo 5350  tpos ctpos 6475

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fo 5358  df-fv 5360  df-tpos 6476

This theorem is referenced by:  tposf2  6499  tposf1o2  6501  tposfo  6502