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Mirrors > Home > MPE Home > Th. List > Mathboxes > nofulllem3 | Structured version Visualization version GIF version |
Description: Lemma for nofull (future) . Restriction of surreal number to a superset of its birthday does not change anything. (Contributed by Scott Fenton, 25-Apr-2017.) |
Ref | Expression |
---|---|
nofulllem3 | ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → (𝑋 ↾ ∪ ( bday “ 𝑆)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3563 | . . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ No ) | |
2 | nofun 31046 | . . . . 5 ⊢ (𝑋 ∈ No → Fun 𝑋) | |
3 | funrel 5821 | . . . . 5 ⊢ (Fun 𝑋 → Rel 𝑋) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑋 ∈ No → Rel 𝑋) |
5 | 1, 4 | syl 17 | . . 3 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → Rel 𝑋) |
6 | 5 | 3adant3 1074 | . 2 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → Rel 𝑋) |
7 | bdayval 31045 | . . . . . 6 ⊢ (𝑋 ∈ No → ( bday ‘𝑋) = dom 𝑋) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → ( bday ‘𝑋) = dom 𝑋) |
9 | bdaydm 31077 | . . . . . . . . 9 ⊢ dom bday = No | |
10 | 1, 9 | syl6eleqr 2699 | . . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom bday ) |
11 | bdayfun 31075 | . . . . . . . 8 ⊢ Fun bday | |
12 | 10, 11 | jctil 558 | . . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → (Fun bday ∧ 𝑋 ∈ dom bday )) |
13 | simpr 476 | . . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
14 | funfvima 6396 | . . . . . . 7 ⊢ ((Fun bday ∧ 𝑋 ∈ dom bday ) → (𝑋 ∈ 𝐴 → ( bday ‘𝑋) ∈ ( bday “ 𝐴))) | |
15 | 12, 13, 14 | sylc 63 | . . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → ( bday ‘𝑋) ∈ ( bday “ 𝐴)) |
16 | elssuni 4403 | . . . . . 6 ⊢ (( bday ‘𝑋) ∈ ( bday “ 𝐴) → ( bday ‘𝑋) ⊆ ∪ ( bday “ 𝐴)) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → ( bday ‘𝑋) ⊆ ∪ ( bday “ 𝐴)) |
18 | 8, 17 | eqsstr3d 3603 | . . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴) → dom 𝑋 ⊆ ∪ ( bday “ 𝐴)) |
19 | 18 | 3adant3 1074 | . . 3 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → dom 𝑋 ⊆ ∪ ( bday “ 𝐴)) |
20 | imass2 5420 | . . . . 5 ⊢ (𝐴 ⊆ 𝑆 → ( bday “ 𝐴) ⊆ ( bday “ 𝑆)) | |
21 | 20 | unissd 4398 | . . . 4 ⊢ (𝐴 ⊆ 𝑆 → ∪ ( bday “ 𝐴) ⊆ ∪ ( bday “ 𝑆)) |
22 | 21 | 3ad2ant3 1077 | . . 3 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → ∪ ( bday “ 𝐴) ⊆ ∪ ( bday “ 𝑆)) |
23 | 19, 22 | sstrd 3578 | . 2 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → dom 𝑋 ⊆ ∪ ( bday “ 𝑆)) |
24 | relssres 5357 | . 2 ⊢ ((Rel 𝑋 ∧ dom 𝑋 ⊆ ∪ ( bday “ 𝑆)) → (𝑋 ↾ ∪ ( bday “ 𝑆)) = 𝑋) | |
25 | 6, 23, 24 | syl2anc 691 | 1 ⊢ ((𝐴 ⊆ No ∧ 𝑋 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑆) → (𝑋 ↾ ∪ ( bday “ 𝑆)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 dom cdm 5038 ↾ cres 5040 “ cima 5041 Rel wrel 5043 Fun wfun 5798 ‘cfv 5804 No csur 31037 bday cbday 31039 |
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-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-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-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-1o 7447 df-no 31040 df-bday 31042 |
This theorem is referenced by: nofulllem4 31104 |
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