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Theorem setrec2lem2 42959
 Description: Lemma for setrec2 42960. 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 5567 . 2 Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2 fvex 6342 . . . . 5 (𝐹𝑥) ∈ V
3 eqeq2 2781 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑦 = 𝑧𝑦 = (𝐹𝑥)))
43imbi2d 329 . . . . . 6 (𝑧 = (𝐹𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
54albidv 2000 . . . . 5 (𝑧 = (𝐹𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
62, 5spcev 3449 . . . 4 (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥)) → ∃𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧))
7 vex 3352 . . . . . 6 𝑦 ∈ V
87brres 5543 . . . . 5 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
9 abid 2758 . . . . . . 7 (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10 tz6.12-1 6351 . . . . . . 7 ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
119, 10sylan2b 573 . . . . . 6 ((𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → (𝐹𝑥) = 𝑦)
1211eqcomd 2776 . . . . 5 ((𝑥𝐹𝑦𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑦 = (𝐹𝑥))
138, 12sylbi 207 . . . 4 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))
146, 13mpg 1871 . . 3 𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
1514ax-gen 1869 . 2 𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
16 nfcv 2912 . . . 4 𝑥𝐹
17 nfab1 2914 . . . 4 𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
1816, 17nfres 5536 . . 3 𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
19 nfcv 2912 . . . 4 𝑦𝐹
20 nfeu1 2627 . . . . 5 𝑦∃!𝑦 𝑥𝐹𝑦
2120nfab 2917 . . . 4 𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
2219, 21nfres 5536 . . 3 𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
23 nfcv 2912 . . 3 𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2418, 22, 23dffun3f 42947 . 2 (Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ↔ (Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ∧ ∀𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)))
251, 15, 24mpbir2an 682 1 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1628   = wceq 1630  ∃wex 1851   ∈ wcel 2144  ∃!weu 2617  {cab 2756   class class class wbr 4784   ↾ cres 5251  Rel wrel 5254  Fun wfun 6025  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039 This theorem is referenced by:  setrec2  42960
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