Theorem marypha2lem1 9319

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
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 4992	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
3		snssi 4757	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		fvssunirn 6853	. . . 4 ⊢ (𝐹‘𝑥) ⊆ ∪ ran 𝐹
5		xpss12 5629	. . . 4 ⊢ (({𝑥} ⊆ 𝐴 ∧ (𝐹‘𝑥) ⊆ ∪ ran 𝐹) → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
6	3, 4, 5	sylancl 586	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
7	2, 6	mprgbir 3054	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)
8	1, 7	eqsstri 3976	1 ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)

Syntax hints:   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  {csn 4573  ∪ cuni 4856  ∪ ciun 4939   × cxp 5612  ran crn 5615  ‘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-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  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-cnv 5622  df-dm 5624  df-rn 5625  df-iota 6437  df-fv 6489

This theorem is referenced by:  marypha2  9323