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Theorem zfrep6 7645
Description: A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5194 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 5181. (Contributed by NM, 10-Oct-2003.)
Assertion
Ref Expression
zfrep6 (∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
Distinct variable groups:   𝜑,𝑤   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem zfrep6
StepHypRef Expression
1 euex 2655 . . . . . . 7 (∃!𝑦𝜑 → ∃𝑦𝜑)
21ralimi 3157 . . . . . 6 (∀𝑥𝑧 ∃!𝑦𝜑 → ∀𝑥𝑧𝑦𝜑)
3 rabid2 3379 . . . . . 6 (𝑧 = {𝑥𝑧 ∣ ∃𝑦𝜑} ↔ ∀𝑥𝑧𝑦𝜑)
42, 3sylibr 235 . . . . 5 (∀𝑥𝑧 ∃!𝑦𝜑𝑧 = {𝑥𝑧 ∣ ∃𝑦𝜑})
5 19.42v 1945 . . . . . . 7 (∃𝑦(𝑥𝑧𝜑) ↔ (𝑥𝑧 ∧ ∃𝑦𝜑))
65abbii 2883 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥𝑧𝜑)} = {𝑥 ∣ (𝑥𝑧 ∧ ∃𝑦𝜑)}
7 dmopab 5777 . . . . . 6 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝑧𝜑)}
8 df-rab 3144 . . . . . 6 {𝑥𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑧 ∧ ∃𝑦𝜑)}
96, 7, 83eqtr4i 2851 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = {𝑥𝑧 ∣ ∃𝑦𝜑}
104, 9syl6reqr 2872 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = 𝑧)
11 vex 3495 . . . 4 𝑧 ∈ V
1210, 11syl6eqel 2918 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∈ V)
13 eumo 2656 . . . . . . 7 (∃!𝑦𝜑 → ∃*𝑦𝜑)
1413imim2i 16 . . . . . 6 ((𝑥𝑧 → ∃!𝑦𝜑) → (𝑥𝑧 → ∃*𝑦𝜑))
15 moanimv 2697 . . . . . 6 (∃*𝑦(𝑥𝑧𝜑) ↔ (𝑥𝑧 → ∃*𝑦𝜑))
1614, 15sylibr 235 . . . . 5 ((𝑥𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥𝑧𝜑))
1716alimi 1803 . . . 4 (∀𝑥(𝑥𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥𝑧𝜑))
18 df-ral 3140 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥𝑧 → ∃!𝑦𝜑))
19 funopab 6383 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝑧𝜑))
2017, 18, 193imtr4i 293 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
21 funrnex 7644 . . 3 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∈ V → (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∈ V))
2212, 20, 21sylc 65 . 2 (∀𝑥𝑧 ∃!𝑦𝜑 → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∈ V)
23 nfra1 3216 . . 3 𝑥𝑥𝑧 ∃!𝑦𝜑
2410eleq2d 2895 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ 𝑥𝑧))
25 opabidw 5403 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ (𝑥𝑧𝜑))
26 vex 3495 . . . . . . . . . 10 𝑥 ∈ V
27 vex 3495 . . . . . . . . . 10 𝑦 ∈ V
2826, 27opelrn 5806 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
2925, 28sylbir 236 . . . . . . . 8 ((𝑥𝑧𝜑) → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
3029ex 413 . . . . . . 7 (𝑥𝑧 → (𝜑𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}))
3130impac 553 . . . . . 6 ((𝑥𝑧𝜑) → (𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
3231eximi 1826 . . . . 5 (∃𝑦(𝑥𝑧𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
337abeq2i 2945 . . . . 5 (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ ∃𝑦(𝑥𝑧𝜑))
34 df-rex 3141 . . . . 5 (∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑 ↔ ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
3532, 33, 343imtr4i 293 . . . 4 (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑)
3624, 35syl6bir 255 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → (𝑥𝑧 → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
3723, 36ralrimi 3213 . 2 (∀𝑥𝑧 ∃!𝑦𝜑 → ∀𝑥𝑧𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑)
38 nfopab1 5126 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
3938nfrn 5817 . . . 4 𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4039nfeq2 2992 . . 3 𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
41 nfcv 2974 . . . 4 𝑦𝑤
42 nfopab2 5127 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4342nfrn 5817 . . . 4 𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4441, 43rexeqf 3396 . . 3 (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → (∃𝑦𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
4540, 44ralbid 3228 . 2 (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → (∀𝑥𝑧𝑦𝑤 𝜑 ↔ ∀𝑥𝑧𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
4622, 37, 45spcedv 3596 1 (∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  ∃!weu 2646  {cab 2796  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cop 4563  {copab 5119  dom cdm 5548  ran crn 5549  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  bnj865  32094
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