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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 2737 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) | |
| 2 | sneq 4636 | . . . 4 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤}) | |
| 3 | 2 | imaeq2d 6078 | . . 3 ⊢ (𝑥 = 𝑤 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑤})) |
| 4 | 3 | inteqd 4951 | . 2 ⊢ (𝑥 = 𝑤 → ∩ (◡𝐹 “ {𝑥}) = ∩ (◡𝐹 “ {𝑤})) |
| 5 | simpr 484 | . 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ 𝐴) | |
| 6 | cnvimass 6100 | . . . 4 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹 | |
| 7 | dnnumch.f | . . . . . 6 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) | |
| 8 | 7 | tfr1 8437 | . . . . 5 ⊢ 𝐹 Fn On |
| 9 | 8 | fndmi 6672 | . . . 4 ⊢ dom 𝐹 = On |
| 10 | 6, 9 | sseqtri 4032 | . . 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 43057 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
| 14 | 13 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹) |
| 15 | inisegn0 6116 | . . . 4 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅) | |
| 16 | 14, 15 | sylib 218 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅) |
| 17 | oninton 7815 | . . 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 7039 | 1 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ∩ cint 4946 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 Oncon0 6384 ‘cfv 6561 recscrecs 8410 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 |
| This theorem is referenced by: dnnumch3 43059 dnwech 43060 |
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