Theorem mpo2eqb 7283

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7281. (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		pm13.183 3659	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
2	1	ralimi 3160	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
3		ralbi 3167	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) → (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
4	2, 3	syl 17	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
5	4	ralimi 3160	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
6		ralbi 3167	. . 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)))
7	5, 6	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
8		df-mpo 7161	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
9		df-mpo 7161	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)}
10	8, 9	eqeq12i 2836	. . 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)})
11		eqoprab2bw 7224	. . 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)))
12		pm5.32 576	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
13	12	albii 1820	. . . . . 6 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
14		19.21v 1940	. . . . . 6 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
15	13, 14	bitr3i 279	. . . . 5 ⊢ (∀𝑧(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
16	15	2albii 1821	. . . 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)))
17		r2al 3201	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
18	16, 17	bitr4i 280	. . 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))
19	10, 11, 18	3bitri 299	. 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷))
20	7, 19	syl6rbbr 292	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   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {coprab 7157   ∈ cmpo 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  homfeq  16964  comfeq  16976