Theorem tposf12 6272

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 5008	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 6228	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴)
4		f1of1 5462	. . . . . . 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 5081	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8	6, 7	sylib 122	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡◡𝐴 = 𝐴)
9		f1eq3 5420	. . . . . . 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 5428	. . . . . . . 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 4805	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡dom 𝐹 = ◡𝐴)
15		mpteq1 4089	. . . . . 6 ⊢ (◡dom 𝐹 = ◡𝐴 → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}))
16		f1eq1 5418	. . . . . 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 5435	. . . 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 4712	. . . . 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 6264	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
24		f1eq1 5418	. . . 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 1353  {csn 3594  ∪ cuni 3811   ↦ cmpt 4066  ^◡ccnv 4627  dom cdm 4628   ∘ ccom 4632  Rel wrel 4633  –1-1→wf1 5215  –1-1-onto→wf1o 5217  tpos ctpos 6247

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-tpos 6248

This theorem is referenced by:  tposf1o2  6273