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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnnumch2 | Structured version Visualization version GIF version |
Description: Define an enumeration (weak dominance version) of a set from a choice 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 |
---|---|
dnnumch2 | ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnnumch.f | . . 3 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) | |
2 | dnnumch.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | dnnumch.g | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) | |
4 | 1, 2, 3 | dnnumch1 36632 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) |
5 | f1ofo 6057 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → (𝐹 ↾ 𝑥):𝑥–onto→𝐴) | |
6 | forn 6031 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴) |
8 | resss 5342 | . . . . . 6 ⊢ (𝐹 ↾ 𝑥) ⊆ 𝐹 | |
9 | rnss 5275 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑥) ⊆ 𝐹 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹) | |
10 | 8, 9 | mp1i 13 | . . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹) |
11 | 7, 10 | eqsstr3d 3603 | . . . 4 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)) |
13 | 12 | rexlimdvw 3016 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)) |
14 | 4, 13 | mpd 15 | 1 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 Oncon0 5640 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 recscrecs 7354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-wrecs 7294 df-recs 7355 |
This theorem is referenced by: dnnumch3lem 36634 dnnumch3 36635 |
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