Theorem tposf12 8201

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 6069	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 8061	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴–1-1-onto→◡◡𝐴)
4		f1of1 6779	. . . . . . 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 6153	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8	6, 7	sylib 218	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡◡𝐴 = 𝐴)
9		f1eq3 6733	. . . . . . 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 6740	. . . . . . . 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 5830	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → ◡dom 𝐹 = ◡𝐴)
15		mpteq1 5174	. . . . . 6 ⊢ (◡dom 𝐹 = ◡𝐴 → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}))
16		f1eq1 6731	. . . . . 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 6747	. . . 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 585	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴–1-1→𝐵)
21	12	releqd 5735	. . . . 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 8193	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
24		f1eq1 6731	. . . 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 1542  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166  ^◡ccnv 5630  dom cdm 5631   ∘ ccom 5635  Rel wrel 5636  –1-1→wf1 6495  –1-1-onto→wf1o 6497  tpos ctpos 8175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1st 7942  df-2nd 7943  df-tpos 8176

This theorem is referenced by:  tposf1o2  8202