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Theorem marypha2lem1 8937
 Description: Lemma for marypha2 8941. 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 4937 . . 3 ( 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹) ↔ ∀𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
3 snssi 4701 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 fvssunirn 6691 . . . 4 (𝐹𝑥) ⊆ ran 𝐹
5 xpss12 5542 . . . 4 (({𝑥} ⊆ 𝐴 ∧ (𝐹𝑥) ⊆ ran 𝐹) → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
63, 4, 5sylancl 589 . . 3 (𝑥𝐴 → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
72, 6mprgbir 3085 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹)
81, 7eqsstri 3928 1 𝑇 ⊆ (𝐴 × ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  {csn 4525  ∪ cuni 4801  ∪ ciun 4886   × cxp 5525  ran crn 5528  ‘cfv 6339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-iota 6298  df-fv 6347 This theorem is referenced by:  marypha2  8941
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