Theorem zfrep6OLD 7898

Description: Obsolete proof of zfrep6 5212 as of 5-Apr-2026. (Contributed by NM, 10-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
zfrep6OLD	⊢ (∀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 zfrep6OLD

Step	Hyp	Ref	Expression
1		19.42v 1960	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑))
2	1	abbii 2806	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
3		dmopab 5858	. . . . . 6 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑)}
4		df-rab 3392	. . . . . 6 ⊢ {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑧 ∧ ∃𝑦𝜑)}
5	2, 3, 4	3eqtr4i 2772	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑}
6		euex 2581	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑)
7	6	ralimi 3076	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
8		rabid2 3424	. . . . . 6 ⊢ (𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑} ↔ ∀𝑥 ∈ 𝑧 ∃𝑦𝜑)
9	7, 8	sylibr 235	. . . . 5 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → 𝑧 = {𝑥 ∈ 𝑧 ∣ ∃𝑦𝜑})
10	5, 9	eqtr4id 2793	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} = 𝑧)
11		vex 3435	. . . 4 ⊢ 𝑧 ∈ V
12	10, 11	eqeltrdi 2847	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∈ V)
13		eumo 2582	. . . . . . 7 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
14	13	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
15		moanimv 2623	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 → ∃*𝑦𝜑))
16	14, 15	sylibr 235	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
17	16	alimi 1818	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑) → ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
18		df-ral 3054	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑧 → ∃!𝑦𝜑))
19		funopab 6521	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
20	17, 18, 19	3imtr4i 293	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
21		funrnex 7897	. . 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 3263	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑧 ∃!𝑦𝜑
24	10	eleq2d 2825	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ 𝑥 ∈ 𝑧))
25		opabidw 5467	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
26		vex 3435	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27		vex 3435	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
28	26, 27	opelrn 5886	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
29	25, 28	sylbir 236	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)})
30	29	ex 413	. . . . . . 7 ⊢ (𝑥 ∈ 𝑧 → (𝜑 → 𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}))
31	30	impac 557	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → (𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
32	31	eximi 1842	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑) → ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
33	3	eqabri 2881	. . . . 5 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ↔ ∃𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
34		df-rex 3064	. . . . 5 ⊢ (∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑 ↔ ∃𝑦(𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} ∧ 𝜑))
35	32, 33, 34	3imtr4i 293	. . . 4 ⊢ (𝑥 ∈ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
36	24, 35	biimtrrdi 255	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
37	23, 36	ralrimi 3237	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑)
38		nfopab1 5143	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
39	38	nfrn 5895	. . . 4 ⊢ Ⅎ𝑥ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
40	39	nfeq2 2918	. . 3 ⊢ Ⅎ𝑥 𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
41		nfcv 2901	. . . 4 ⊢ Ⅎ𝑦𝑤
42		nfopab2 5144	. . . . 5 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
43	42	nfrn 5895	. . . 4 ⊢ Ⅎ𝑦ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}
44	41, 43	rexeqf 3321	. . 3 ⊢ (𝑤 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)} → (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦 ∈ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑧 ∧ 𝜑)}𝜑))
45	40, 44	ralbid 3252	. 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 3536	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃*wmo 2541  ∃!weu 2572  {cab 2717  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431  ⟨cop 4562  {copab 5135  dom cdm 5619  ran crn 5620  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by: (None)