Theorem mpo2eqb 5951

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5949. (Contributed by Mario Carneiro, 4-Jan-2017.)

Assertion

Ref	Expression
mpo2eqb	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem mpo2eqb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 5847	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		df-mpo 5847	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)}
3	1, 2	eqeq12i 2179	. . 3 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)})
4		eqoprab2b 5900	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)} ↔ ∀𝑥∀𝑦∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
5		pm5.32 449	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
6	5	albii 1458	. . . . . 6 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
7		19.21v 1861	. . . . . 6 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
8	6, 7	bitr3i 185	. . . . 5 ⊢ (∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
9	8	2albii 1459	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
10		r2al 2485	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
11	9, 10	bitr4i 186	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷))
12	3, 4, 11	3bitri 205	. 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷))
13		pm13.183 2864	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
14	13	ralimi 2529	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
15		ralbi 2598	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) → (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
16	14, 15	syl 14	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
17	16	ralimi 2529	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
18		ralbi 2598	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
19	17, 18	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
20	12, 19	bitr4id 198	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {coprab 5843   ∈ cmpo 5844

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-in1 604  ax-in2 605  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  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  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-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-oprab 5846  df-mpo 5847

This theorem is referenced by: (None)