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Theorem zfrep6 7667
 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 5174 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 5161. (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 19.42v 1955 . . . . . . 7 (∃𝑦(𝑥𝑧𝜑) ↔ (𝑥𝑧 ∧ ∃𝑦𝜑))
21abbii 2824 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥𝑧𝜑)} = {𝑥 ∣ (𝑥𝑧 ∧ ∃𝑦𝜑)}
3 dmopab 5762 . . . . . 6 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝑧𝜑)}
4 df-rab 3080 . . . . . 6 {𝑥𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥𝑧 ∧ ∃𝑦𝜑)}
52, 3, 43eqtr4i 2792 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = {𝑥𝑧 ∣ ∃𝑦𝜑}
6 euex 2597 . . . . . . 7 (∃!𝑦𝜑 → ∃𝑦𝜑)
76ralimi 3093 . . . . . 6 (∀𝑥𝑧 ∃!𝑦𝜑 → ∀𝑥𝑧𝑦𝜑)
8 rabid2 3300 . . . . . 6 (𝑧 = {𝑥𝑧 ∣ ∃𝑦𝜑} ↔ ∀𝑥𝑧𝑦𝜑)
97, 8sylibr 237 . . . . 5 (∀𝑥𝑧 ∃!𝑦𝜑𝑧 = {𝑥𝑧 ∣ ∃𝑦𝜑})
105, 9eqtr4id 2813 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} = 𝑧)
11 vex 3414 . . . 4 𝑧 ∈ V
1210, 11eqeltrdi 2861 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∈ V)
13 eumo 2598 . . . . . . 7 (∃!𝑦𝜑 → ∃*𝑦𝜑)
1413imim2i 16 . . . . . 6 ((𝑥𝑧 → ∃!𝑦𝜑) → (𝑥𝑧 → ∃*𝑦𝜑))
15 moanimv 2641 . . . . . 6 (∃*𝑦(𝑥𝑧𝜑) ↔ (𝑥𝑧 → ∃*𝑦𝜑))
1614, 15sylibr 237 . . . . 5 ((𝑥𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥𝑧𝜑))
1716alimi 1814 . . . 4 (∀𝑥(𝑥𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥𝑧𝜑))
18 df-ral 3076 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥𝑧 → ∃!𝑦𝜑))
19 funopab 6376 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝑧𝜑))
2017, 18, 193imtr4i 295 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
21 funrnex 7666 . . 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 3148 . . 3 𝑥𝑥𝑧 ∃!𝑦𝜑
2410eleq2d 2838 . . . 4 (∀𝑥𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ 𝑥𝑧))
25 opabidw 5387 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ (𝑥𝑧𝜑))
26 vex 3414 . . . . . . . . . 10 𝑥 ∈ V
27 vex 3414 . . . . . . . . . 10 𝑦 ∈ V
2826, 27opelrn 5790 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
2925, 28sylbir 238 . . . . . . . 8 ((𝑥𝑧𝜑) → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)})
3029ex 416 . . . . . . 7 (𝑥𝑧 → (𝜑𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}))
3130impac 556 . . . . . 6 ((𝑥𝑧𝜑) → (𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
3231eximi 1837 . . . . 5 (∃𝑦(𝑥𝑧𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
333abeq2i 2888 . . . . 5 (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ↔ ∃𝑦(𝑥𝑧𝜑))
34 df-rex 3077 . . . . 5 (∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑 ↔ ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} ∧ 𝜑))
3532, 33, 343imtr4i 295 . . . 4 (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑)
3624, 35syl6bir 257 . . 3 (∀𝑥𝑧 ∃!𝑦𝜑 → (𝑥𝑧 → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
3723, 36ralrimi 3145 . 2 (∀𝑥𝑧 ∃!𝑦𝜑 → ∀𝑥𝑧𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑)
38 nfopab1 5106 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
3938nfrn 5799 . . . 4 𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4039nfeq2 2937 . . 3 𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
41 nfcv 2920 . . . 4 𝑦𝑤
42 nfopab2 5107 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4342nfrn 5799 . . . 4 𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}
4441, 43rexeqf 3317 . . 3 (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → (∃𝑦𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
4540, 44ralbid 3160 . 2 (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)} → (∀𝑥𝑧𝑦𝑤 𝜑 ↔ ∀𝑥𝑧𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑧𝜑)}𝜑))
4622, 37, 45spcedv 3520 1 (∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∃*wmo 2556  ∃!weu 2588  {cab 2736  ∀wral 3071  ∃wrex 3072  {crab 3075  Vcvv 3410  ⟨cop 4532  {copab 5099  dom cdm 5529  ran crn 5530  Fun wfun 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349 This theorem is referenced by:  bnj865  32437
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