Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dnnumch3lem | Structured version Visualization version GIF version |
Description: Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
dnnumch.f | ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) |
dnnumch.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
dnnumch.g | ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) |
Ref | Expression |
---|---|
dnnumch3lem | ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) | |
2 | sneq 4567 | . . . 4 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤}) | |
3 | 2 | imaeq2d 5922 | . . 3 ⊢ (𝑥 = 𝑤 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑤})) |
4 | 3 | inteqd 4872 | . 2 ⊢ (𝑥 = 𝑤 → ∩ (◡𝐹 “ {𝑥}) = ∩ (◡𝐹 “ {𝑤})) |
5 | simpr 485 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴) | |
6 | cnvimass 5942 | . . . 4 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹 | |
7 | dnnumch.f | . . . . . 6 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) | |
8 | 7 | tfr1 8022 | . . . . 5 ⊢ 𝐹 Fn On |
9 | fndm 6448 | . . . . 5 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ dom 𝐹 = On |
11 | 6, 10 | sseqtri 4000 | . . 3 ⊢ (◡𝐹 “ {𝑤}) ⊆ On |
12 | dnnumch.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | dnnumch.g | . . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) | |
14 | 7, 12, 13 | dnnumch2 39523 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
15 | 14 | sselda 3964 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹) |
16 | inisegn0 5954 | . . . 4 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅) | |
17 | 15, 16 | sylib 219 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅) |
18 | oninton 7504 | . . 3 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ On) | |
19 | 11, 17, 18 | sylancr 587 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ On) |
20 | 1, 4, 5, 19 | fvmptd3 6783 | 1 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 ∩ cint 4867 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 Oncon0 6184 Fn wfn 6343 ‘cfv 6348 recscrecs 7996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-wrecs 7936 df-recs 7997 |
This theorem is referenced by: dnnumch3 39525 dnwech 39526 |
Copyright terms: Public domain | W3C validator |