Mathbox for Stefan O'Rear |
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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 | simpr 477 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴) | |
2 | cnvimass 5485 | . . . 4 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹 | |
3 | dnnumch.f | . . . . . 6 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) | |
4 | 3 | tfr1 7493 | . . . . 5 ⊢ 𝐹 Fn On |
5 | fndm 5990 | . . . . 5 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ dom 𝐹 = On |
7 | 2, 6 | sseqtri 3637 | . . 3 ⊢ (◡𝐹 “ {𝑤}) ⊆ On |
8 | dnnumch.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | dnnumch.g | . . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) | |
10 | 3, 8, 9 | dnnumch2 37615 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
11 | 10 | sselda 3603 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹) |
12 | inisegn0 5497 | . . . 4 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅) | |
13 | 11, 12 | sylib 208 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅) |
14 | oninton 7000 | . . 3 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ On) | |
15 | 7, 13, 14 | sylancr 695 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ On) |
16 | sneq 4187 | . . . . 5 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤}) | |
17 | 16 | imaeq2d 5466 | . . . 4 ⊢ (𝑥 = 𝑤 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑤})) |
18 | 17 | inteqd 4480 | . . 3 ⊢ (𝑥 = 𝑤 → ∩ (◡𝐹 “ {𝑥}) = ∩ (◡𝐹 “ {𝑤})) |
19 | eqid 2622 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) | |
20 | 18, 19 | fvmptg 6280 | . 2 ⊢ ((𝑤 ∈ 𝐴 ∧ ∩ (◡𝐹 “ {𝑤}) ∈ On) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
21 | 1, 15, 20 | syl2anc 693 | 1 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∩ cint 4475 ↦ cmpt 4729 ◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Oncon0 5723 Fn wfn 5883 ‘cfv 5888 recscrecs 7467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-wrecs 7407 df-recs 7468 |
This theorem is referenced by: dnnumch3 37617 dnwech 37618 |
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