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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 2738 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) | |
2 | sneq 4571 | . . . 4 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤}) | |
3 | 2 | imaeq2d 5969 | . . 3 ⊢ (𝑥 = 𝑤 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑤})) |
4 | 3 | inteqd 4884 | . 2 ⊢ (𝑥 = 𝑤 → ∩ (◡𝐹 “ {𝑥}) = ∩ (◡𝐹 “ {𝑤})) |
5 | simpr 485 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴) | |
6 | cnvimass 5989 | . . . 4 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹 | |
7 | dnnumch.f | . . . . . 6 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) | |
8 | 7 | tfr1 8228 | . . . . 5 ⊢ 𝐹 Fn On |
9 | 8 | fndmi 6537 | . . . 4 ⊢ dom 𝐹 = On |
10 | 6, 9 | sseqtri 3957 | . . 3 ⊢ (◡𝐹 “ {𝑤}) ⊆ On |
11 | dnnumch.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | dnnumch.g | . . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) | |
13 | 7, 11, 12 | dnnumch2 40870 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
14 | 13 | sselda 3921 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹) |
15 | inisegn0 6006 | . . . 4 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅) | |
16 | 14, 15 | sylib 217 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅) |
17 | oninton 7645 | . . 3 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ On) | |
18 | 10, 16, 17 | sylancr 587 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ On) |
19 | 1, 4, 5, 18 | fvmptd3 6898 | 1 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ∩ cint 4879 ↦ cmpt 5157 ◡ccnv 5588 dom cdm 5589 ran crn 5590 “ cima 5592 Oncon0 6266 ‘cfv 6433 recscrecs 8201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 |
This theorem is referenced by: dnnumch3 40872 dnwech 40873 |
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