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Theorem wfaxrep 44949
Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5284. 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 44936 . . . 4 Tr (𝑅1 “ On)
3 treq 5272 . . . 4 (𝑊 = (𝑅1 “ On) → (Tr 𝑊 ↔ Tr (𝑅1 “ On)))
42, 3mpbiri 258 . . 3 (𝑊 = (𝑅1 “ On) → Tr 𝑊)
5 vex 3481 . . . . . . . . . 10 𝑓 ∈ V
65rnex 7932 . . . . . . . . 9 ran 𝑓 ∈ V
76r1elss 9843 . . . . . . . 8 (ran 𝑓 (𝑅1 “ On) ↔ ran 𝑓 (𝑅1 “ On))
87biimpri 228 . . . . . . 7 (ran 𝑓 (𝑅1 “ On) → ran 𝑓 (𝑅1 “ On))
91sseq2i 4024 . . . . . . 7 (ran 𝑓𝑊 ↔ ran 𝑓 (𝑅1 “ On))
101eleq2i 2830 . . . . . . 7 (ran 𝑓𝑊 ↔ ran 𝑓 (𝑅1 “ On))
118, 9, 103imtr4i 292 . . . . . 6 (ran 𝑓𝑊 → ran 𝑓𝑊)
12113ad2ant3 1134 . . . . 5 ((Fun 𝑓 ∧ dom 𝑓𝑊 ∧ ran 𝑓𝑊) → ran 𝑓𝑊)
1312ax-gen 1791 . . . 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 9867 . . . . 5 On ⊆ (𝑅1 “ On)
16 0elon 6439 . . . . 5 ∅ ∈ On
1715, 16sselii 3991 . . . 4 ∅ ∈ (𝑅1 “ On)
18 eleq2 2827 . . . 4 (𝑊 = (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ (𝑅1 “ On)))
1917, 18mpbiri 258 . . 3 (𝑊 = (𝑅1 “ On) → ∅ ∈ 𝑊)
204, 14, 19modelaxrep 44945 . 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 1086  wal 1534   = wceq 1536  wcel 2105  wral 3058  wrex 3067  wss 3962  c0 4338   cuni 4911  Tr wtr 5264  dom cdm 5688  ran crn 5689  cima 5691  Oncon0 6385  Fun wfun 6556  𝑅1cr1 9799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-en 8984  df-dom 8985  df-sdom 8986  df-r1 9801  df-rank 9802
This theorem is referenced by: (None)
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