Theorem tposf12 8234

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 6096	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 8094	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴)
4		f1of1 6825	. . . . . . 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 6181	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8	6, 7	sylib 217	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡◡𝐴 = 𝐴)
9		f1eq3 6777	. . . . . . 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 232	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1→𝐴)
12		f1dm 6784	. . . . . . . 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 5868	. . . . . 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 6775	. . . . . 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 6792	. . . 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 583	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵)
21	12	releqd 5771	. . . . 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 8226	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
24		f1eq1 6775	. . . 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 205   ∧ wa 395   = wceq 1533  {csn 4623  ∪ cuni 4902   ↦ cmpt 5224  ^◡ccnv 5668  dom cdm 5669   ∘ ccom 5673  Rel wrel 5674  –1-1→wf1 6533  –1-1-onto→wf1o 6535  tpos ctpos 8208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7971  df-2nd 7972  df-tpos 8209

This theorem is referenced by:  tposf1o2  8235