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Theorem wfaxrep 44987
Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5277. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in 𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.)
Hypothesis
Ref Expression
wfax.1 𝑊 = (𝑅1 “ On)
Assertion
Ref Expression
wfaxrep 𝑥𝑊 (∀𝑤𝑊𝑦𝑊𝑧𝑊 (∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑊𝑧𝑊 (𝑧𝑦 ↔ ∃𝑤𝑊 (𝑤𝑥 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑊
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wfaxrep
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 wfax.1 . 2 𝑊 = (𝑅1 “ On)
2 trwf 44954 . . . 4 Tr (𝑅1 “ On)
3 treq 5265 . . . 4 (𝑊 = (𝑅1 “ On) → (Tr 𝑊 ↔ Tr (𝑅1 “ On)))
42, 3mpbiri 258 . . 3 (𝑊 = (𝑅1 “ On) → Tr 𝑊)
5 vex 3483 . . . . . . . . . 10 𝑓 ∈ V
65rnex 7928 . . . . . . . . 9 ran 𝑓 ∈ V
76r1elss 9842 . . . . . . . 8 (ran 𝑓 (𝑅1 “ On) ↔ ran 𝑓 (𝑅1 “ On))
87biimpri 228 . . . . . . 7 (ran 𝑓 (𝑅1 “ On) → ran 𝑓 (𝑅1 “ On))
91sseq2i 4012 . . . . . . 7 (ran 𝑓𝑊 ↔ ran 𝑓 (𝑅1 “ On))
101eleq2i 2832 . . . . . . 7 (ran 𝑓𝑊 ↔ ran 𝑓 (𝑅1 “ On))
118, 9, 103imtr4i 292 . . . . . 6 (ran 𝑓𝑊 → ran 𝑓𝑊)
12113ad2ant3 1136 . . . . 5 ((Fun 𝑓 ∧ dom 𝑓𝑊 ∧ ran 𝑓𝑊) → ran 𝑓𝑊)
1312ax-gen 1795 . . . 4 𝑓((Fun 𝑓 ∧ dom 𝑓𝑊 ∧ ran 𝑓𝑊) → ran 𝑓𝑊)
1413a1i 11 . . 3 (𝑊 = (𝑅1 “ On) → ∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑊 ∧ ran 𝑓𝑊) → ran 𝑓𝑊))
15 onwf 9866 . . . . 5 On ⊆ (𝑅1 “ On)
16 0elon 6436 . . . . 5 ∅ ∈ On
1715, 16sselii 3979 . . . 4 ∅ ∈ (𝑅1 “ On)
18 eleq2 2829 . . . 4 (𝑊 = (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ (𝑅1 “ On)))
1917, 18mpbiri 258 . . 3 (𝑊 = (𝑅1 “ On) → ∅ ∈ 𝑊)
204, 14, 19modelaxrep 44974 . 2 (𝑊 = (𝑅1 “ On) → ∀𝑥𝑊 (∀𝑤𝑊𝑦𝑊𝑧𝑊 (∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑊𝑧𝑊 (𝑧𝑦 ↔ ∃𝑤𝑊 (𝑤𝑥 ∧ ∀𝑦𝜑))))
211, 20ax-mp 5 1 𝑥𝑊 (∀𝑤𝑊𝑦𝑊𝑧𝑊 (∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑊𝑧𝑊 (𝑧𝑦 ↔ ∃𝑤𝑊 (𝑤𝑥 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wral 3060  wrex 3069  wss 3950  c0 4332   cuni 4905  Tr wtr 5257  dom cdm 5683  ran crn 5684  cima 5686  Oncon0 6382  Fun wfun 6553  𝑅1cr1 9798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-om 7884  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-en 8982  df-dom 8983  df-sdom 8984  df-r1 9800  df-rank 9801
This theorem is referenced by: (None)
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