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Theorem marypha2lem4 9476
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
marypha2lem4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑋
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
21marypha2lem2 9474 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
32imaeq1i 6077 . . . 4 (𝑇𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋)
4 df-ima 5702 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
53, 4eqtri 2763 . . 3 (𝑇𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
6 resopab2 6056 . . . . . 6 (𝑋𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
76adantl 481 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
87rneqd 5952 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
9 rnopab 5968 . . . . 5 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))}
10 df-rex 3069 . . . . . . . . 9 (∃𝑥𝑋 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)))
1110bicomi 224 . . . . . . . 8 (∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥))
1211abbii 2807 . . . . . . 7 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
13 df-iun 4998 . . . . . . 7 𝑥𝑋 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
1412, 13eqtr4i 2766 . . . . . 6 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥)
1514a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
169, 15eqtrid 2787 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
178, 16eqtrd 2775 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = 𝑥𝑋 (𝐹𝑥))
185, 17eqtrid 2787 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = 𝑥𝑋 (𝐹𝑥))
19 fnfun 6669 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2019adantr 480 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → Fun 𝐹)
21 funiunfv 7268 . . 3 (Fun 𝐹 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2220, 21syl 17 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2318, 22eqtrd 2775 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  wss 3963  {csn 4631   cuni 4912   ciun 4996  {copab 5210   × cxp 5687  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557   Fn wfn 6558  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-in 3970  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-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-fv 6571
This theorem is referenced by:  marypha2  9477
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