Theorem marypha2lem1 9350

Description: Lemma for marypha2 9354. 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
marypha2lem1	⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem1

Step	Hyp	Ref	Expression
1		marypha2lem.t	. 2 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))
2		iunss 5002	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
3		snssi 4766	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		fvssunirn 6873	. . . 4 ⊢ (𝐹‘𝑥) ⊆ ∪ ran 𝐹
5		xpss12 5647	. . . 4 ⊢ (({𝑥} ⊆ 𝐴 ∧ (𝐹‘𝑥) ⊆ ∪ ran 𝐹) → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
6	3, 4, 5	sylancl 587	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
7	2, 6	mprgbir 3059	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)
8	1, 7	eqsstri 3982	1 ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)

Syntax hints:   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  {csn 4582  ∪ cuni 4865  ∪ ciun 4948   × cxp 5630  ran crn 5633  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-iota 6456  df-fv 6508

This theorem is referenced by:  marypha2  9354