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Theorem marypha2lem4 9478
Description: Lemma for marypha2 9479. 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 9476 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
32imaeq1i 6075 . . . 4 (𝑇𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋)
4 df-ima 5698 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
53, 4eqtri 2765 . . 3 (𝑇𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
6 resopab2 6054 . . . . . 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 5949 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
9 rnopab 5965 . . . . 5 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))}
10 df-rex 3071 . . . . . . . . 9 (∃𝑥𝑋 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)))
1110bicomi 224 . . . . . . . 8 (∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥))
1211abbii 2809 . . . . . . 7 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
13 df-iun 4993 . . . . . . 7 𝑥𝑋 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
1412, 13eqtr4i 2768 . . . . . 6 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥)
1514a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
169, 15eqtrid 2789 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
178, 16eqtrd 2777 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = 𝑥𝑋 (𝐹𝑥))
185, 17eqtrid 2789 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = 𝑥𝑋 (𝐹𝑥))
19 fnfun 6668 . . . 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 2777 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  wss 3951  {csn 4626   cuni 4907   ciun 4991  {copab 5205   × cxp 5683  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  marypha2  9479
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