Theorem mptcnv 5993

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 5125	. 2 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)})
3		df-mpt 5140	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 5740	. . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 5992	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2844	. 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7		df-mpt 5140	. 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}
8	2, 6, 7	3eqtr4g 2881	1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {copab 5121   ↦ cmpt 5139  ^◡ccnv 5549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-mpt 5140  df-xp 5556  df-rel 5557  df-cnv 5558

This theorem is referenced by:  nvocnv  7032  mptfzshft  15127  fprodrev  15325  pt1hmeo  22408  ballotlemrinv  31786  dssmapnvod  40359