Theorem zfrep6 7027

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 4709 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 4699. (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		euex 2482	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
2	1	ralimi 2936	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
3		rabid2 3096	. . . . . 6 ⊢ (𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} ↔ ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
4	2, 3	sylibr 223	. . . . 5 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → 𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑})
5		19.42v 1905	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑))
6	5	abbii 2726	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
7		dmopab 5257	. . . . . 6 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)}
8		df-rab 2905	. . . . . 6 ⊢ {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
9	6, 7, 8	3eqtr4i 2642	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑}
10	4, 9	syl6reqr 2663	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = 𝑧)
11		vex 3176	. . . 4 ⊢ 𝑧 ∈ V
12	10, 11	syl6eqel 2696	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
13		eumo 2487	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
14	13	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
15		moanimv 2519	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
16	14, 15	sylibr 223	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
17	16	alimi 1730	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
18		df-ral 2901	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
19		funopab 5837	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
20	17, 18, 19	3imtr4i 280	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
21		funrnex 7026	. . 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 63	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
23		nfra1 2925	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑧 ∃!𝑦𝜑
24	10	eleq2d 2673	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ 𝑥 ∈ 𝑧))
25		opabid 4907	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
26		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27		vex 3176	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
28	26, 27	opelrn 5278	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
29	25, 28	sylbir 224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
30	29	ex 449	. . . . . . 7 ⊢ (𝑥 ∈ 𝑧 → (𝜑 → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}))
31	30	impac 649	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → (𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
32	31	eximi 1752	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
33	7	abeq2i 2722	. . . . 5 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
34		df-rex 2902	. . . . 5 ⊢ (∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑 ↔ ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
35	32, 33, 34	3imtr4i 280	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
36	24, 35	syl6bir 243	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
37	23, 36	ralrimi 2940	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
38		nfopab1 4651	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
39	38	nfrn 5289	. . . . 5 ⊢ Ⅎ𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
40	39	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
41		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦𝑤
42		nfopab2 4652	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
43	42	nfrn 5289	. . . . 5 ⊢ Ⅎ𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
44	41, 43	rexeqf 3112	. . . 4 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
45	40, 44	ralbid 2966	. . 3 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑 ↔ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
46	45	spcegv 3267	. 2 ⊢ (ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
47	22, 37, 46	sylc 63	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cop 4131  {copab 4642  dom cdm 5038  ran crn 5039  Fun wfun 5798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  bnj865  30247