Theorem tposf12 8277

Description: Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposf12	⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵))

Proof of Theorem tposf12

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		relcnv 6121	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 8137	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴)
4		f1of1 6846	. . . . . . 7 ⊢ ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴)
5	2, 3, 4	mp2b 10	. . . . . 6 ⊢ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴
6		simpl 482	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → Rel 𝐴)
7		dfrel2 6208	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8	6, 7	sylib 218	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡◡𝐴 = 𝐴)
9		f1eq3 6800	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴))
10	8, 9	syl 17	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴))
11	5, 10	mpbii 233	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴)
12		f1dm 6807	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
13	1, 12	syl 17	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
14	13	cnveqd 5885	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡dom 𝐹 = ◡𝐴)
15		mpteq1 5234	. . . . . 6 ⊢ (◡dom 𝐹 = ◡𝐴 → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}))
16		f1eq1 6798	. . . . . 6 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}) → ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴))
17	14, 15, 16	3syl 18	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴))
18	11, 17	mpbird 257	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴)
19		f1co 6814	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵)
20	1, 18, 19	syl2anc 584	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵)
21	12	releqd 5787	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (Rel dom 𝐹 ↔ Rel 𝐴))
22	21	biimparc 479	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → Rel dom 𝐹)
23		dftpos2 8269	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
24		f1eq1 6798	. . . 4 ⊢ (tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) → (tpos 𝐹:◡𝐴–1-1→𝐵 ↔ (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵))
25	22, 23, 24	3syl 18	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (tpos 𝐹:◡𝐴–1-1→𝐵 ↔ (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵))
26	20, 25	mpbird 257	. 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → tpos 𝐹:◡𝐴–1-1→𝐵)
27	26	ex 412	1 ⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224  ^◡ccnv 5683  dom cdm 5684   ∘ ccom 5688  Rel wrel 5689  –1-1→wf1 6557  –1-1-onto→wf1o 6559  tpos ctpos 8251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-1st 8015  df-2nd 8016  df-tpos 8252

This theorem is referenced by:  tposf1o2  8278