Theorem tposf12 8296

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

Assertion

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

Proof of Theorem tposf12

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹:𝐴⟶1-1𝐵)
2		relcnv 6112	. . . . . . 7 ⊢ Rel ◡𝐴
3		cnvf1o 8149	. . . . . . 7 ⊢ (Rel ◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1onto◡◡𝐴)
4		f1of1 6845	. . . . . . 7 ⊢ ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1onto◡◡𝐴 → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1◡◡𝐴)
5	2, 3, 4	mp2b 10	. . . . . 6 ⊢ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1◡◡𝐴
6		dfrel2 6193	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
7	6	birani 509	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → ◡◡𝐴 = 𝐴)
8		f1eq3 6797	. . . . . . 7 ⊢ (◡◡𝐴 = 𝐴 → ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1◡◡𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴))
9	7, 8	syl 18	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → ((𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1◡◡𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴))
10	5, 9	mpbii 236	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴)
11		f1dm 6806	. . . . . . . 8 ⊢ (𝐹:𝐴⟶1-1𝐵 → dom 𝐹 = 𝐴)
12	1, 11	syl 18	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → dom 𝐹 = 𝐴)
13	12	cnveqd 5860	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → ◡dom 𝐹 = ◡𝐴)
14		mpteq1 5197	. . . . . 6 ⊢ (◡dom 𝐹 = ◡𝐴 → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}))
15		f1eq1 6795	. . . . . 6 ⊢ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}) → ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴))
16	13, 14, 15	3syl 19	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴 ↔ (𝑥 ∈ ◡𝐴 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴))
17	10, 16	mpbird 260	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴)
18		f1co 6813	. . . 4 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}):◡𝐴⟶1-1𝐴) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴⟶1-1𝐵)
19	1, 17, 18	syl2anc 596	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴⟶1-1𝐵)
20	11	releqd 5760	. . . . 5 ⊢ (𝐹:𝐴⟶1-1𝐵 → (Rel dom 𝐹 ↔ Rel 𝐴))
21	20	biimparc 485	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → Rel dom 𝐹)
22		dftpos2 8288	. . . 4 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))
23		f1eq1 6795	. . . 4 ⊢ (tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) → (tpos 𝐹:◡𝐴⟶1-1𝐵 ↔ (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴⟶1-1𝐵))
24	21, 22, 23	3syl 19	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → (tpos 𝐹:◡𝐴⟶1-1𝐵 ↔ (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})):◡𝐴⟶1-1𝐵))
25	19, 24	mpbird 260	. 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶1-1𝐵) → tpos 𝐹:◡𝐴⟶1-1𝐵)
26	25	ex 418	1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶1-1𝐵 → tpos 𝐹:◡𝐴⟶1-1𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189  ^◡ccnv 5654  dom cdm 5655   ∘ ccom 5659  Rel wrel 5660  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557  tpos ctpos 8270

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-1st 8025  df-2nd 8026  df-tpos 8271

This theorem is used by:  tposf1o2  8297