Theorem tz9.12lem1 9200

Description: Lemma for tz9.12 9203. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
tz9.12lem.1	⊢ 𝐴 ∈ V
tz9.12lem.2	⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})

Assertion

Ref	Expression
tz9.12lem1	⊢ (𝐹 “ 𝐴) ⊆ On

Distinct variable group:   𝑧,𝑣,𝐴

Allowed substitution hints:   𝐹(𝑧,𝑣)

Proof of Theorem tz9.12lem1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5907	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
2		tz9.12lem.2	. . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
3	2	rnmpt 5791	. . 3 ⊢ ran 𝐹 = {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}}
4		id 22	. . . . . 6 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
5		ssrab2 4007	. . . . . . 7 ⊢ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On
6		eqvisset 3458	. . . . . . . 8 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
7		intex 5204	. . . . . . . 8 ⊢ ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
8	6, 7	sylibr 237	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅)
9		oninton 7495	. . . . . . 7 ⊢ (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅) → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
10	5, 8, 9	sylancr 590	. . . . . 6 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
11	4, 10	eqeltrd 2890	. . . . 5 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
12	11	rexlimivw 3241	. . . 4 ⊢ (∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
13	12	abssi 3997	. . 3 ⊢ {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}} ⊆ On
14	3, 13	eqsstri 3949	. 2 ⊢ ran 𝐹 ⊆ On
15	1, 14	sstri 3924	1 ⊢ (𝐹 “ 𝐴) ⊆ On

Syntax hints:   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838   ↦ cmpt 5110  ran crn 5520   “ cima 5522  Oncon0 6159  ‘cfv 6324  𝑅₁cr1 9175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163

This theorem is referenced by:  tz9.12lem2  9201  tz9.12lem3  9202