Theorem tposf12 6426

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 110	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		relcnv 5109	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 6382	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴)
4		f1of1 5576	. . . . . . 7 ⊢ ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴)
5	2, 3, 4	mp2b 8	. . . . . 6 ⊢ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴
6		simpl 109	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → Rel 𝐴)
7		dfrel2 5182	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8	6, 7	sylib 122	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡◡𝐴 = 𝐴)
9		f1eq3 5533	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→◡◡𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴))
10	8, 9	syl 14	. . . . . 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 148	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴)
12		f1dm 5541	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
13	1, 12	syl 14	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → dom 𝐹 = 𝐴)
14	13	cnveqd 4901	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡dom 𝐹 = ◡𝐴)
15		mpteq1 4168	. . . . . 6 ⊢ (◡dom 𝐹 = ◡𝐴 → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}))
16		f1eq1 5531	. . . . . 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 17	. . . . 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 167	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴)
19		f1co 5548	. . . 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 411	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵)
21	12	releqd 4805	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (Rel dom 𝐹 ↔ Rel 𝐴))
22	21	biimparc 299	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → Rel dom 𝐹)
23		dftpos2 6418	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
24		f1eq1 5531	. . . 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 17	. . 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 167	. 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → tpos 𝐹:◡𝐴–1-1→𝐵)
27	26	ex 115	1 ⊢ (Rel 𝐴 → (𝐹:𝐴–1-1→𝐵 → tpos 𝐹:◡𝐴–1-1→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  {csn 3666  ∪ cuni 3888   ↦ cmpt 4145  ^◡ccnv 4719  dom cdm 4720   ∘ ccom 4724  Rel wrel 4725  –1-1→wf1 5318  –1-1-onto→wf1o 5320  tpos ctpos 6401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-1st 6295  df-2nd 6296  df-tpos 6402

This theorem is referenced by:  tposf1o2  6427