Theorem tz9.12lem1 9735

Description: Lemma for tz9.12 9738. (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 6050	. 2 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
2		tz9.12lem.2	. . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
3	2	rnmpt 5926	. . 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 4028	. . . . . . 7 ⊢ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On
6		eqvisset 3468	. . . . . . . 8 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
7		intex 5294	. . . . . . . 8 ⊢ ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V)
8	6, 7	sylibr 236	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅)
9		oninton 7767	. . . . . . 7 ⊢ (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅) → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
10	5, 8, 9	sylancr 595	. . . . . 6 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On)
11	4, 10	eqeltrd 2856	. . . . 5 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
12	11	rexlimivw 3153	. . . 4 ⊢ (∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On)
13	12	abssi 4016	. . 3 ⊢ {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}} ⊆ On
14	3, 13	eqsstri 3977	. 2 ⊢ ran 𝐹 ⊆ On
15	1, 14	sstri 3940	1 ⊢ (𝐹 “ 𝐴) ⊆ On

Syntax hints:   = wceq 1554   ∈ wcel 2136  {cab 2734   ≠ wne 2951  ∃wrex 3080  {crab 3408  Vcvv 3448   ⊆ wss 3899  ∅c0 4280  ∩ cint 4899   ↦ cmpt 5175  ran crn 5641   “ cima 5643  Oncon0 6335  ‘cfv 6510  𝑅₁cr1 9710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ord 6338  df-on 6339

This theorem is referenced by:  tz9.12lem2  9736  tz9.12lem3  9737