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Mirrors > Home > MPE Home > Th. List > tz9.12lem1 | Structured version Visualization version GIF version |
Description: Lemma for tz9.12 9213. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
tz9.12lem.1 | ⊢ 𝐴 ∈ V |
tz9.12lem.2 | ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) |
Ref | Expression |
---|---|
tz9.12lem1 | ⊢ (𝐹 “ 𝐴) ⊆ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 5934 | . 2 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
2 | tz9.12lem.2 | . . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) | |
3 | 2 | rnmpt 5821 | . . 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 4055 | . . . . . . 7 ⊢ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On | |
6 | eqvisset 3511 | . . . . . . . 8 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V) | |
7 | intex 5232 | . . . . . . . 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 7509 | . . . . . . 7 ⊢ (({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅) → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On) | |
10 | 5, 8, 9 | sylancr 589 | . . . . . 6 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ On) |
11 | 4, 10 | eqeltrd 2913 | . . . . 5 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On) |
12 | 11 | rexlimivw 3282 | . . . 4 ⊢ (∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On) |
13 | 12 | abssi 4045 | . . 3 ⊢ {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}} ⊆ On |
14 | 3, 13 | eqsstri 4000 | . 2 ⊢ ran 𝐹 ⊆ On |
15 | 1, 14 | sstri 3975 | 1 ⊢ (𝐹 “ 𝐴) ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ∩ cint 4868 ↦ cmpt 5138 ran crn 5550 “ cima 5552 Oncon0 6185 ‘cfv 6349 𝑅1cr1 9185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 |
This theorem is referenced by: tz9.12lem2 9211 tz9.12lem3 9212 |
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