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Theorem tposfo2 5792
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 5791 . . . 4 (Rel A → (𝐹 Fn A → tpos 𝐹 Fn A))
21adantrd 264 . . 3 (Rel A → ((𝐹 Fn A ran 𝐹 = B) → tpos 𝐹 Fn A))
3 fndm 4912 . . . . . . . . 9 (𝐹 Fn A → dom 𝐹 = A)
43releqd 4339 . . . . . . . 8 (𝐹 Fn A → (Rel dom 𝐹 ↔ Rel A))
54biimparc 283 . . . . . . 7 ((Rel A 𝐹 Fn A) → Rel dom 𝐹)
6 rntpos 5782 . . . . . . 7 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
75, 6syl 14 . . . . . 6 ((Rel A 𝐹 Fn A) → ran tpos 𝐹 = ran 𝐹)
87eqeq1d 2021 . . . . 5 ((Rel A 𝐹 Fn A) → (ran tpos 𝐹 = B ↔ ran 𝐹 = B))
98biimprd 147 . . . 4 ((Rel A 𝐹 Fn A) → (ran 𝐹 = B → ran tpos 𝐹 = B))
109expimpd 345 . . 3 (Rel A → ((𝐹 Fn A ran 𝐹 = B) → ran tpos 𝐹 = B))
112, 10jcad 291 . 2 (Rel A → ((𝐹 Fn A ran 𝐹 = B) → (tpos 𝐹 Fn A ran tpos 𝐹 = B)))
12 df-fo 4823 . 2 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
13 df-fo 4823 . 2 (tpos 𝐹:AontoB ↔ (tpos 𝐹 Fn A ran tpos 𝐹 = B))
1411, 12, 133imtr4g 194 1 (Rel A → (𝐹:AontoB → tpos 𝐹:AontoB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223  ccnv 4259  dom cdm 4260  ran crn 4261  Rel wrel 4265   Fn wfn 4812  ontowfo 4815  tpos ctpos 5769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-nul 3846  ax-pow 3890  ax-pr 3907  ax-un 4108
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-sbc 2733  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-fo 4823  df-fv 4825  df-tpos 5770
This theorem is referenced by:  tposf2  5793  tposf1o2  5795  tposfo  5796
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