Theorem marypha2lem2 8695

Description: Lemma for marypha2 8698. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypothesis

Ref	Expression
marypha2lem.t	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))

Assertion

Ref	Expression
marypha2lem2	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}

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

Allowed substitution hints:   𝑇(𝑥,𝑦)

Proof of Theorem marypha2lem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		marypha2lem.t	. 2 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
2		sneq 4451	. . . 4 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
3		fveq2 6499	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
4	2, 3	xpeq12d 5438	. . 3 ⊢ (𝑥 = 𝑧 → ({𝑥} × (𝐹‘𝑥)) = ({𝑧} × (𝐹‘𝑧)))
5	4	cbviunv 4833	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧))
6		df-xp 5413	. . . . 5 ⊢ ({𝑧} × (𝐹‘𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
7	6	a1i 11	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ({𝑧} × (𝐹‘𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))})
8	7	iuneq2i 4812	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
9		iunopab 5298	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
10		velsn 4457	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11		equcom 1975	. . . . . . . 8 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
12	10, 11	bitri 267	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
13	12	anbi1i 614	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
14	13	rexbii 3195	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
15		fveq2 6499	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
16	15	eleq2d 2852	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ (𝐹‘𝑧) ↔ 𝑦 ∈ (𝐹‘𝑥)))
17	16	ceqsrexbv 3565	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
18	14, 17	bitri 267	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
19	18	opabbii 4996	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
20	8, 9, 19	3eqtri 2807	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
21	1, 5, 20	3eqtri 2807	1 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}

Syntax hints:   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3090  {csn 4441  ∪ ciun 4792  {copab 4991   × cxp 5405  ‘cfv 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-xp 5413  df-iota 6152  df-fv 6196

This theorem is referenced by:  marypha2lem3  8696  marypha2lem4  8697  eulerpartlemgu  31277