Theorem mptcnv 5006

Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Hypothesis

Ref	Expression
mptcnv.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)))

Assertion

Ref	Expression
mptcnv	⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷))

Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝐶   𝑥,𝐷   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)

Proof of Theorem mptcnv

Step	Hyp	Ref	Expression
1		mptcnv.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)))
2	1	opabbidv 4048	. 2 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)})
3		df-mpt 4045	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 4779	. . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 5005	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2186	. 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7		df-mpt 4045	. 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}
8	2, 6, 7	3eqtr4g 2224	1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {copab 4042   ↦ cmpt 4043  ^◡ccnv 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by: (None)