Theorem marypha2lem2 9320

Description: Lemma for marypha2 9323. 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 4583	. . . 4 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
3		fveq2 6822	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
4	2, 3	xpeq12d 5645	. . 3 ⊢ (𝑥 = 𝑧 → ({𝑥} × (𝐹‘𝑥)) = ({𝑧} × (𝐹‘𝑧)))
5	4	cbviunv 4987	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) = ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧))
6		df-xp 5620	. . . . 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 4961	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
9		iunopab 5497	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))}
10		velsn 4589	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11		equcom 2019	. . . . . . . 8 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
12	10, 11	bitri 275	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
13	12	anbi1i 624	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
14	13	rexbii 3079	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)))
15		fveq2 6822	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
16	15	eleq2d 2817	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ (𝐹‘𝑧) ↔ 𝑦 ∈ (𝐹‘𝑥)))
17	16	ceqsrexbv 3606	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 (𝑧 = 𝑥 ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
18	14, 17	bitri 275	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))
19	18	opabbii 5156	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹‘𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
20	8, 9, 19	3eqtri 2758	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 ({𝑧} × (𝐹‘𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
21	1, 5, 20	3eqtri 2758	1 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {csn 4573  ∪ ciun 4939  {copab 5151   × cxp 5612  ‘cfv 6481

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-xp 5620  df-iota 6437  df-fv 6489

This theorem is referenced by:  marypha2lem3  9321  marypha2lem4  9322  eulerpartlemgu  34390