Theorem zfrep6 7933

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 5251 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 5234. (Contributed by NM, 10-Oct-2003.)

Assertion

Ref	Expression
zfrep6	⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Distinct variable groups:   𝜑,𝑤   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑))
2	1	abbii 2796	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
3		dmopab 5879	. . . . . 6 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)}
4		df-rab 3406	. . . . . 6 ⊢ {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
5	2, 3, 4	3eqtr4i 2762	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑}
6		euex 2570	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
7	6	ralimi 3066	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
8		rabid2 3439	. . . . . 6 ⊢ (𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} ↔ ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
9	7, 8	sylibr 234	. . . . 5 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → 𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑})
10	5, 9	eqtr4id 2783	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = 𝑧)
11		vex 3451	. . . 4 ⊢ 𝑧 ∈ V
12	10, 11	eqeltrdi 2836	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
13		eumo 2571	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
14	13	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
15		moanimv 2612	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
16	14, 15	sylibr 234	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
17	16	alimi 1811	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
18		df-ral 3045	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
19		funopab 6551	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
20	17, 18, 19	3imtr4i 292	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
21		funrnex 7932	. . 3 ⊢ (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V → (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V))
22	12, 20, 21	sylc 65	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
23		nfra1 3261	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑧 ∃!𝑦𝜑
24	10	eleq2d 2814	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ 𝑥 ∈ 𝑧))
25		opabidw 5484	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
26		vex 3451	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27		vex 3451	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
28	26, 27	opelrn 5907	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
29	25, 28	sylbir 235	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
30	29	ex 412	. . . . . . 7 ⊢ (𝑥 ∈ 𝑧 → (𝜑 → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}))
31	30	impac 552	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → (𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
32	31	eximi 1835	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
33	3	eqabri 2871	. . . . 5 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
34		df-rex 3054	. . . . 5 ⊢ (∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑 ↔ ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
35	32, 33, 34	3imtr4i 292	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
36	24, 35	biimtrrdi 254	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
37	23, 36	ralrimi 3235	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
38		nfopab1 5177	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
39	38	nfrn 5916	. . . 4 ⊢ Ⅎ𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
40	39	nfeq2 2909	. . 3 ⊢ Ⅎ𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
41		nfcv 2891	. . . 4 ⊢ Ⅎ𝑦𝑤
42		nfopab2 5178	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
43	42	nfrn 5916	. . . 4 ⊢ Ⅎ𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
44	41, 43	rexeqf 3330	. . 3 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
45	40, 44	ralbid 3250	. 2 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑 ↔ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
46	22, 37, 45	spcedv 3564	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  {cab 2707  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447  ⟨cop 4595  {copab 5169  dom cdm 5638  ran crn 5639  Fun wfun 6505

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  bnj865  34913