Theorem mpo2eqb 6120

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6118. (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 6012	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		df-mpo 6012	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)}
3	1, 2	eqeq12i 2243	. . 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 6068	. . 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		pm5.32 453	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐷)))
6	5	albii 1516	. . . . . 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 1919	. . . . . 6 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
8	6, 7	bitr3i 186	. . . . 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 1517	. . . 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 2549	. . . 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 187	. . 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 206	. 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷))
13		pm13.183 2941	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
14	13	ralimi 2593	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
15		ralbi 2663	. . . . 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 2593	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝐶 = 𝐷 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧(𝑧 = 𝐶 ↔ 𝑧 = 𝐷)))
18		ralbi 2663	. . 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 199	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 104   ↔ wb 105  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {coprab 6008   ∈ cmpo 6009

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-oprab 6011  df-mpo 6012

This theorem is referenced by: (None)