Theorem marypha2lem2 9195

Description: Lemma for marypha2 9198. 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 4571	. . . 4 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
3		fveq2 6774	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
4	2, 3	xpeq12d 5620	. . 3 ⊢ (𝑥 = 𝑧 → ({𝑥} × (𝐹‘𝑥)) = ({𝑧} × (𝐹‘𝑧)))
5	4	cbviunv 4970	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧))
6		df-xp 5595	. . . . 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 4945	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
9		iunopab 5472	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
10		velsn 4577	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11		equcom 2021	. . . . . . . 8 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
12	10, 11	bitri 274	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
13	12	anbi1i 624	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
14	13	rexbii 3181	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
15		fveq2 6774	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
16	15	eleq2d 2824	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ (𝐹‘𝑧) ↔ 𝑦 ∈ (𝐹‘𝑥)))
17	16	ceqsrexbv 3586	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
18	14, 17	bitri 274	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
19	18	opabbii 5141	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
20	8, 9, 19	3eqtri 2770	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
21	1, 5, 20	3eqtri 2770	1 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {csn 4561  ∪ ciun 4924  {copab 5136   × cxp 5587  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-iota 6391  df-fv 6441

This theorem is referenced by:  marypha2lem3  9196  marypha2lem4  9197  eulerpartlemgu  32344