Theorem zfrep6 7995

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 5317 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 5303. (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 2812	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
3		dmopab 5940	. . . . . 6 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)}
4		df-rab 3444	. . . . . 6 ⊢ {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
5	2, 3, 4	3eqtr4i 2778	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑}
6		euex 2580	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
7	6	ralimi 3089	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
8		rabid2 3478	. . . . . 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 2799	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = 𝑧)
11		vex 3492	. . . 4 ⊢ 𝑧 ∈ V
12	10, 11	eqeltrdi 2852	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
13		eumo 2581	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
14	13	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
15		moanimv 2622	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
16	14, 15	sylibr 234	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
17	16	alimi 1809	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
18		df-ral 3068	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
19		funopab 6613	. . . 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 7994	. . 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 3290	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑧 ∃!𝑦𝜑
24	10	eleq2d 2830	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ 𝑥 ∈ 𝑧))
25		opabidw 5543	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
26		vex 3492	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27		vex 3492	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
28	26, 27	opelrn 5968	. . . . . . . . 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 1833	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
33	3	eqabri 2888	. . . . 5 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
34		df-rex 3077	. . . . 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 3263	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
38		nfopab1 5236	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
39	38	nfrn 5977	. . . 4 ⊢ Ⅎ𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
40	39	nfeq2 2926	. . 3 ⊢ Ⅎ𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
41		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦𝑤
42		nfopab2 5237	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
43	42	nfrn 5977	. . . 4 ⊢ Ⅎ𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
44	41, 43	rexeqf 3362	. . 3 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
45	40, 44	ralbid 3279	. 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 3611	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571  {cab 2717  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488  ⟨cop 4654  {copab 5228  dom cdm 5700  ran crn 5701  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  bnj865  34899