Theorem tz9.12lem1 9747

Description: Lemma for tz9.12 9750. (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 6045	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
2		tz9.12lem.2	. . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
3	2	rnmpt 5924	. . 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 4046	. . . . . . 7 ⊢ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On
6		eqvisset 3470	. . . . . . . 8 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
7		intex 5302	. . . . . . . 8 ⊢ ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
8	6, 7	sylibr 234	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅)
9		oninton 7774	. . . . . . 7 ⊢ (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅) → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
10	5, 8, 9	sylancr 587	. . . . . 6 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
11	4, 10	eqeltrd 2829	. . . . 5 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
12	11	rexlimivw 3131	. . . 4 ⊢ (∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
13	12	abssi 4036	. . 3 ⊢ {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}} ⊆ On
14	3, 13	eqsstri 3996	. 2 ⊢ ran 𝐹 ⊆ On
15	1, 14	sstri 3959	1 ⊢ (𝐹 “ 𝐴) ⊆ On

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∃wrex 3054  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ∩ cint 4913   ↦ cmpt 5191  ran crn 5642   “ cima 5644  Oncon0 6335  ‘cfv 6514  𝑅₁cr1 9722

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339

This theorem is referenced by:  tz9.12lem2  9748  tz9.12lem3  9749