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Mirrors > Home > MPE Home > Th. List > Mathboxes > elno2 | Structured version Visualization version GIF version |
Description: An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012.) |
Ref | Expression |
---|---|
elno2 | ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nofun 31046 | . . 3 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
2 | nodmon 31047 | . . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On) | |
3 | norn 31048 | . . 3 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜}) | |
4 | 1, 2, 3 | 3jca 1235 | . 2 ⊢ (𝐴 ∈ No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
5 | simp2 1055 | . . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → dom 𝐴 ∈ On) | |
6 | simpl 472 | . . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴) | |
7 | eqidd 2611 | . . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → dom 𝐴 = dom 𝐴) | |
8 | df-fn 5807 | . . . . . . . 8 ⊢ (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) | |
9 | 6, 7, 8 | sylanbrc 695 | . . . . . . 7 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴) |
10 | 9 | anim1i 590 | . . . . . 6 ⊢ (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
11 | 10 | 3impa 1251 | . . . . 5 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
12 | df-f 5808 | . . . . 5 ⊢ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) | |
13 | 11, 12 | sylibr 223 | . . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}) |
14 | feq2 5940 | . . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1𝑜, 2𝑜} ↔ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜})) | |
15 | 14 | rspcev 3282 | . . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜}) |
16 | 5, 13, 15 | syl2anc 691 | . . 3 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜}) |
17 | elno 31043 | . . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜}) | |
18 | 16, 17 | sylibr 223 | . 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴 ∈ No ) |
19 | 4, 18 | impbii 198 | 1 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 {cpr 4127 dom cdm 5038 ran crn 5039 Oncon0 5640 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 1𝑜c1o 7440 2𝑜c2o 7441 No csur 31037 |
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-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-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-no 31040 |
This theorem is referenced by: elno3 31052 |
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