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Theorem brwdom3i 9621
Description: Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
Assertion
Ref Expression
brwdom3i (𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦))
Distinct variable groups:   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Proof of Theorem brwdom3i
StepHypRef Expression
1 relwdom 9604 . . . 4 Rel ≼*
21brrelex1i 5745 . . 3 (𝑋* 𝑌𝑋 ∈ V)
31brrelex2i 5746 . . 3 (𝑋* 𝑌𝑌 ∈ V)
4 brwdom3 9620 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
52, 3, 4syl2anc 584 . 2 (𝑋* 𝑌 → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
65ibi 267 1 (𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478   class class class wbr 5148  cfv 6563  * cwdom 9602
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-pow 5371  ax-pr 5438  ax-un 7754
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-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-en 8985  df-dom 8986  df-sdom 8987  df-wdom 9603
This theorem is referenced by:  unwdomg  9622  xpwdomg  9623
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