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Theorem tposfo2 7640
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo2 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))

Proof of Theorem tposfo2
StepHypRef Expression
1 tposfn2 7639 . . . 4 (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))
21adantrd 487 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn 𝐴))
3 fndm 6223 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43releqd 5438 . . . . . . . 8 (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
54biimparc 473 . . . . . . 7 ((Rel 𝐴𝐹 Fn 𝐴) → Rel dom 𝐹)
6 rntpos 7630 . . . . . . 7 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
75, 6syl 17 . . . . . 6 ((Rel 𝐴𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
87eqeq1d 2827 . . . . 5 ((Rel 𝐴𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
98biimprd 240 . . . 4 ((Rel 𝐴𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
109expimpd 447 . . 3 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
112, 10jcad 510 . 2 (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12 df-fo 6129 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13 df-fo 6129 . 2 (tpos 𝐹:𝐴onto𝐵 ↔ (tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵))
1411, 12, 133imtr4g 288 1 (Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  ccnv 5341  dom cdm 5342  ran crn 5343  Rel wrel 5347   Fn wfn 6118  ontowfo 6121  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-fo 6129  df-fv 6131  df-tpos 7617
This theorem is referenced by:  tposf2  7641  tposf1o2  7643  tposfo  7644  oppglsm  18408
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