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Theorem marypha2lem4 8329
Description: Lemma for marypha2 8330. 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 8327 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
32imaeq1i 5451 . . . 4 (𝑇𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋)
4 df-ima 5117 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
53, 4eqtri 2642 . . 3 (𝑇𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
6 resopab2 5436 . . . . . 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 5342 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
9 rnopab 5359 . . . . 5 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))}
10 df-rex 2915 . . . . . . . . 9 (∃𝑥𝑋 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)))
1110bicomi 214 . . . . . . . 8 (∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥))
1211abbii 2737 . . . . . . 7 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
13 df-iun 4513 . . . . . . 7 𝑥𝑋 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
1412, 13eqtr4i 2645 . . . . . 6 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥)
1514a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
169, 15syl5eq 2666 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
178, 16eqtrd 2654 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = 𝑥𝑋 (𝐹𝑥))
185, 17syl5eq 2666 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = 𝑥𝑋 (𝐹𝑥))
19 fnfun 5976 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2019adantr 481 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → Fun 𝐹)
21 funiunfv 6491 . . 3 (Fun 𝐹 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2220, 21syl 17 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2318, 22eqtrd 2654 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wrex 2910  wss 3567  {csn 4168   cuni 4427   ciun 4511  {copab 4703   × cxp 5102  ran crn 5105  cres 5106  cima 5107  Fun wfun 5870   Fn wfn 5871  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  marypha2  8330
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