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Theorem marypha2lem4 8288
Description: Lemma for marypha2 8289. 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 8286 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
32imaeq1i 5422 . . . 4 (𝑇𝑋) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋)
4 df-ima 5087 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} “ 𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
53, 4eqtri 2643 . . 3 (𝑇𝑋) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋)
6 resopab2 5407 . . . . . 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 5313 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))})
9 rnopab 5330 . . . . 5 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))}
10 df-rex 2913 . . . . . . . . 9 (∃𝑥𝑋 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)))
1110bicomi 214 . . . . . . . 8 (∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥)) ↔ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥))
1211abbii 2736 . . . . . . 7 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
13 df-iun 4487 . . . . . . 7 𝑥𝑋 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝑋 𝑦 ∈ (𝐹𝑥)}
1412, 13eqtr4i 2646 . . . . . 6 {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥)
1514a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → {𝑦 ∣ ∃𝑥(𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
169, 15syl5eq 2667 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦 ∈ (𝐹𝑥))} = 𝑥𝑋 (𝐹𝑥))
178, 16eqtrd 2655 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↾ 𝑋) = 𝑥𝑋 (𝐹𝑥))
185, 17syl5eq 2667 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = 𝑥𝑋 (𝐹𝑥))
19 fnfun 5946 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2019adantr 481 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → Fun 𝐹)
21 funiunfv 6460 . . 3 (Fun 𝐹 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2220, 21syl 17 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑥𝑋 (𝐹𝑥) = (𝐹𝑋))
2318, 22eqtrd 2655 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  wss 3555  {csn 4148   cuni 4402   ciun 4485  {copab 4672   × cxp 5072  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  marypha2  8289
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