Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tz9.12lem1 | Structured version Visualization version GIF version |
Description: Lemma for tz9.12 9647. (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 6010 | . 2 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
2 | tz9.12lem.2 | . . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) | |
3 | 2 | rnmpt 5896 | . . 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 4025 | . . . . . . 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 5281 | . . . . . . . 8 ⊢ ({𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V) | |
8 | 6, 7 | sylibr 233 | . . . . . . 7 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ≠ ∅) |
9 | oninton 7708 | . . . . . . 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 2837 | . . . . 5 ⊢ (𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On) |
12 | 11 | rexlimivw 3144 | . . . 4 ⊢ (∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ On) |
13 | 12 | abssi 4015 | . . 3 ⊢ {𝑥 ∣ ∃𝑧 ∈ V 𝑥 = ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}} ⊆ On |
14 | 3, 13 | eqsstri 3966 | . 2 ⊢ ran 𝐹 ⊆ On |
15 | 1, 14 | sstri 3941 | 1 ⊢ (𝐹 “ 𝐴) ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {cab 2713 ≠ wne 2940 ∃wrex 3070 {crab 3403 Vcvv 3441 ⊆ wss 3898 ∅c0 4269 ∩ cint 4894 ↦ cmpt 5175 ran crn 5621 “ cima 5623 Oncon0 6302 ‘cfv 6479 𝑅1cr1 9619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-cnv 5628 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 |
This theorem is referenced by: tz9.12lem2 9645 tz9.12lem3 9646 |
Copyright terms: Public domain | W3C validator |