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Theorem marypha2lem1 8301
Description: Lemma for marypha2 8305. 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 4534 . . 3 ( 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹) ↔ ∀𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
3 snssi 4315 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 fvssunirn 6184 . . . 4 (𝐹𝑥) ⊆ ran 𝐹
5 xpss12 5196 . . . 4 (({𝑥} ⊆ 𝐴 ∧ (𝐹𝑥) ⊆ ran 𝐹) → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
63, 4, 5sylancl 693 . . 3 (𝑥𝐴 → ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹))
72, 6mprgbir 2923 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) ⊆ (𝐴 × ran 𝐹)
81, 7eqsstri 3620 1 𝑇 ⊆ (𝐴 × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wss 3560  {csn 4155   cuni 4409   ciun 4492   × cxp 5082  ran crn 5085  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-iota 5820  df-fv 5865
This theorem is referenced by:  marypha2  8305
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