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Theorem marypha2lem1 9476
Description: Lemma for marypha2 9480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypothesis
Ref Expression
marypha2lem.t 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
Assertion
Ref Expression
marypha2lem1 𝑇 ⊆ (𝐴 × ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem1
StepHypRef Expression
1 marypha2lem.t . 2 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
2 iunss 5044 . . 3 ( 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹) ↔ ∀𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
3 snssi 4807 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 fvssunirn 6938 . . . 4 (𝐹𝑥) ⊆ ran 𝐹
5 xpss12 5699 . . . 4 (({𝑥} ⊆ 𝐴 ∧ (𝐹𝑥) ⊆ ran 𝐹) → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
63, 4, 5sylancl 586 . . 3 (𝑥𝐴 → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
72, 6mprgbir 3067 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹)
81, 7eqsstri 4029 1 𝑇 ⊆ (𝐴 × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wss 3950  {csn 4625   cuni 4906   ciun 4990   × cxp 5682  ran crn 5685  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-iota 6513  df-fv 6568
This theorem is referenced by:  marypha2  9480
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