Theorem mptcnv 5170

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 4181	. 2 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)})
3		df-mpt 4178	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 4935	. . 3 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 5169	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2255	. 2 ⊢ ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7		df-mpt 4178	. 2 ⊢ (𝑦 ∈ 𝐶 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)}
8	2, 6, 7	3eqtr4g 2292	1 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {copab 4175   ↦ cmpt 4176  ^◡ccnv 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-cnv 4762

This theorem is referenced by:  ballotfilemrinv  13221