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Theorem marypha2lem1 9473
Description: Lemma for marypha2 9477. 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 5050 . . 3 ( 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹) ↔ ∀𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
3 snssi 4813 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 fvssunirn 6940 . . . 4 (𝐹𝑥) ⊆ ran 𝐹
5 xpss12 5704 . . . 4 (({𝑥} ⊆ 𝐴 ∧ (𝐹𝑥) ⊆ ran 𝐹) → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
63, 4, 5sylancl 586 . . 3 (𝑥𝐴 → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
72, 6mprgbir 3066 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹)
81, 7eqsstri 4030 1 𝑇 ⊆ (𝐴 × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wss 3963  {csn 4631   cuni 4912   ciun 4996   × cxp 5687  ran crn 5690  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571
This theorem is referenced by:  marypha2  9477
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