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Theorem marypha2lem4 9435
Description: Lemma for marypha2 9436. 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 9433 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
32imaeq1i 6056 . . . 4 (𝑇𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋)
4 df-ima 5689 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
53, 4eqtri 2760 . . 3 (𝑇𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
6 resopab2 6036 . . . . . 6 (𝑋𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
76adantl 482 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
87rneqd 5937 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
9 rnopab 5953 . . . . 5 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))}
10 df-rex 3071 . . . . . . . . 9 (∃𝑥𝑋 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)))
1110bicomi 223 . . . . . . . 8 (∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥))
1211abbii 2802 . . . . . . 7 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
13 df-iun 4999 . . . . . . 7 𝑥𝑋 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
1412, 13eqtr4i 2763 . . . . . 6 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥)
1514a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
169, 15eqtrid 2784 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
178, 16eqtrd 2772 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = 𝑥𝑋 (𝐹𝑥))
185, 17eqtrid 2784 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = 𝑥𝑋 (𝐹𝑥))
19 fnfun 6649 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2019adantr 481 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → Fun 𝐹)
21 funiunfv 7249 . . 3 (Fun 𝐹 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2220, 21syl 17 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2318, 22eqtrd 2772 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wrex 3070  wss 3948  {csn 4628   cuni 4908   ciun 4997  {copab 5210   × cxp 5674  ran crn 5677  cres 5678  cima 5679  Fun wfun 6537   Fn wfn 6538  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  marypha2  9436
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