Theorem tposfo2 8246

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 8245	. . . 4 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴))
2	1	adantrd 496	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn ◡𝐴))
3		fndm 6640	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	releqd 5767	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴))
5	4	biimparc 484	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → Rel dom 𝐹)
6		rntpos 8236	. . . . . . 7 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
7	5, 6	syl 18	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹)
8	7	eqeq1d 2765	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵))
9	8	biimprd 251	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵))
10	9	expimpd 458	. . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵))
11	2, 10	jcad 521	. 2 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵)))
12		df-fo 6544	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
13		df-fo 6544	. 2 ⊢ (tpos 𝐹:◡𝐴–onto→𝐵 ↔ (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵))
14	11, 12, 13	3imtr4g 299	1 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  Rel wrel 5668   Fn wfn 6533  –onto→wfo 6536  tpos ctpos 8222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fo 6544  df-fv 6546  df-tpos 8223

This theorem is referenced by:  tposf2  8247  tposf1o2  8249  tposfo  8250  oppglsm  19713