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Theorem tposf12 7264
 Description: Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposf12 (Rel 𝐴 → (𝐹:𝐴1-1𝐵 → tpos 𝐹:𝐴1-1𝐵))

Proof of Theorem tposf12
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . 4 ((Rel 𝐴𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1𝐵)
2 relcnv 5422 . . . . . . 7 Rel 𝐴
3 cnvf1o 7163 . . . . . . 7 (Rel 𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴)
4 f1of1 6049 . . . . . . 7 ((𝑥𝐴 {𝑥}):𝐴1-1-onto𝐴 → (𝑥𝐴 {𝑥}):𝐴1-1𝐴)
52, 3, 4mp2b 10 . . . . . 6 (𝑥𝐴 {𝑥}):𝐴1-1𝐴
6 simpl 472 . . . . . . . 8 ((Rel 𝐴𝐹:𝐴1-1𝐵) → Rel 𝐴)
7 dfrel2 5502 . . . . . . . 8 (Rel 𝐴𝐴 = 𝐴)
86, 7sylib 207 . . . . . . 7 ((Rel 𝐴𝐹:𝐴1-1𝐵) → 𝐴 = 𝐴)
9 f1eq3 6011 . . . . . . 7 (𝐴 = 𝐴 → ((𝑥𝐴 {𝑥}):𝐴1-1𝐴 ↔ (𝑥𝐴 {𝑥}):𝐴1-1𝐴))
108, 9syl 17 . . . . . 6 ((Rel 𝐴𝐹:𝐴1-1𝐵) → ((𝑥𝐴 {𝑥}):𝐴1-1𝐴 ↔ (𝑥𝐴 {𝑥}):𝐴1-1𝐴))
115, 10mpbii 222 . . . . 5 ((Rel 𝐴𝐹:𝐴1-1𝐵) → (𝑥𝐴 {𝑥}):𝐴1-1𝐴)
12 f1dm 6018 . . . . . . . 8 (𝐹:𝐴1-1𝐵 → dom 𝐹 = 𝐴)
131, 12syl 17 . . . . . . 7 ((Rel 𝐴𝐹:𝐴1-1𝐵) → dom 𝐹 = 𝐴)
1413cnveqd 5220 . . . . . 6 ((Rel 𝐴𝐹:𝐴1-1𝐵) → dom 𝐹 = 𝐴)
15 mpteq1 4665 . . . . . 6 (dom 𝐹 = 𝐴 → (𝑥dom 𝐹 {𝑥}) = (𝑥𝐴 {𝑥}))
16 f1eq1 6009 . . . . . 6 ((𝑥dom 𝐹 {𝑥}) = (𝑥𝐴 {𝑥}) → ((𝑥dom 𝐹 {𝑥}):𝐴1-1𝐴 ↔ (𝑥𝐴 {𝑥}):𝐴1-1𝐴))
1714, 15, 163syl 18 . . . . 5 ((Rel 𝐴𝐹:𝐴1-1𝐵) → ((𝑥dom 𝐹 {𝑥}):𝐴1-1𝐴 ↔ (𝑥𝐴 {𝑥}):𝐴1-1𝐴))
1811, 17mpbird 246 . . . 4 ((Rel 𝐴𝐹:𝐴1-1𝐵) → (𝑥dom 𝐹 {𝑥}):𝐴1-1𝐴)
19 f1co 6023 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝑥dom 𝐹 {𝑥}):𝐴1-1𝐴) → (𝐹 ∘ (𝑥dom 𝐹 {𝑥})):𝐴1-1𝐵)
201, 18, 19syl2anc 691 . . 3 ((Rel 𝐴𝐹:𝐴1-1𝐵) → (𝐹 ∘ (𝑥dom 𝐹 {𝑥})):𝐴1-1𝐵)
2112releqd 5126 . . . . 5 (𝐹:𝐴1-1𝐵 → (Rel dom 𝐹 ↔ Rel 𝐴))
2221biimparc 503 . . . 4 ((Rel 𝐴𝐹:𝐴1-1𝐵) → Rel dom 𝐹)
23 dftpos2 7256 . . . 4 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
24 f1eq1 6009 . . . 4 (tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})) → (tpos 𝐹:𝐴1-1𝐵 ↔ (𝐹 ∘ (𝑥dom 𝐹 {𝑥})):𝐴1-1𝐵))
2522, 23, 243syl 18 . . 3 ((Rel 𝐴𝐹:𝐴1-1𝐵) → (tpos 𝐹:𝐴1-1𝐵 ↔ (𝐹 ∘ (𝑥dom 𝐹 {𝑥})):𝐴1-1𝐵))
2620, 25mpbird 246 . 2 ((Rel 𝐴𝐹:𝐴1-1𝐵) → tpos 𝐹:𝐴1-1𝐵)
2726ex 449 1 (Rel 𝐴 → (𝐹:𝐴1-1𝐵 → tpos 𝐹:𝐴1-1𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043  –1-1→wf1 5801  –1-1-onto→wf1o 5803  tpos ctpos 7238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-tpos 7239 This theorem is referenced by:  tposf1o2  7265
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