Theorem marypha2lem1 9338

Description: Lemma for marypha2 9342. 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 4974	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
3		snssi 4717	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		fvssunirn 6858	. . . 4 ⊢ (𝐹‘𝑥) ⊆ ∪ ran 𝐹
5		xpss12 5633	. . . 4 ⊢ (({𝑥} ⊆ 𝐴 ∧ (𝐹‘𝑥) ⊆ ∪ ran 𝐹) → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
6	3, 4, 5	sylancl 592	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹))
7	2, 6	mprgbir 3060	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)
8	1, 7	eqsstri 3961	1 ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)

Syntax hints:   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  {csn 4555  ∪ cuni 4838  ∪ ciun 4921   × cxp 5616  ran crn 5619  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-iota 6441  df-fv 6493

This theorem is referenced by:  marypha2  9342