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Theorem setrec2lem2 42212
Description: Lemma for setrec2 42213. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
Assertion
Ref Expression
setrec2lem2 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem setrec2lem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relres 5424 . 2 Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2 fvex 6199 . . . . 5 (𝐹𝑥) ∈ V
3 eqeq2 2632 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑦 = 𝑧𝑦 = (𝐹𝑥)))
43imbi2d 330 . . . . . 6 (𝑧 = (𝐹𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
54albidv 1848 . . . . 5 (𝑧 = (𝐹𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
62, 5spcev 3298 . . . 4 (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥)) → ∃𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧))
7 vex 3201 . . . . . 6 𝑦 ∈ V
87brres 5400 . . . . 5 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
9 abid 2609 . . . . . . 7 (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10 tz6.12-1 6208 . . . . . . 7 ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
119, 10sylan2b 492 . . . . . 6 ((𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → (𝐹𝑥) = 𝑦)
1211eqcomd 2627 . . . . 5 ((𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑦 = (𝐹𝑥))
138, 12sylbi 207 . . . 4 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))
146, 13mpg 1723 . . 3 𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
1514ax-gen 1721 . 2 𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
16 nfcv 2763 . . . 4 𝑥𝐹
17 nfab1 2765 . . . 4 𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
1816, 17nfres 5396 . . 3 𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
19 nfcv 2763 . . . 4 𝑦𝐹
20 nfeu1 2479 . . . . 5 𝑦∃!𝑦 𝑥𝐹𝑦
2120nfab 2768 . . . 4 𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
2219, 21nfres 5396 . . 3 𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
23 nfcv 2763 . . 3 𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2418, 22, 23dffun3f 42200 . 2 (Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ↔ (Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ∧ ∀𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)))
251, 15, 24mpbir2an 955 1 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480   = wceq 1482  wex 1703  wcel 1989  ∃!weu 2469  {cab 2607   class class class wbr 4651  cres 5114  Rel wrel 5117  Fun wfun 5880  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-res 5124  df-iota 5849  df-fun 5888  df-fv 5894
This theorem is referenced by:  setrec2  42213
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