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Theorem marypha2lem1 9379
Description: Lemma for marypha2 9383. 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 5009 . . 3 ( 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹) ↔ ∀𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
3 snssi 4772 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 fvssunirn 6879 . . . 4 (𝐹𝑥) ⊆ ran 𝐹
5 xpss12 5652 . . . 4 (({𝑥} ⊆ 𝐴 ∧ (𝐹𝑥) ⊆ ran 𝐹) → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
63, 4, 5sylancl 587 . . 3 (𝑥𝐴 → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
72, 6mprgbir 3068 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹)
81, 7eqsstri 3982 1 𝑇 ⊆ (𝐴 × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wss 3914  {csn 4590   cuni 4869   ciun 4958   × cxp 5635  ran crn 5638  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-iota 6452  df-fv 6508
This theorem is referenced by:  marypha2  9383
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