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Theorem nofulllem3 31103
 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.)
Assertion
Ref Expression
nofulllem3 ((𝐴 No 𝑋𝐴𝐴𝑆) → (𝑋 ( bday 𝑆)) = 𝑋)

Proof of Theorem nofulllem3
StepHypRef Expression
1 ssel2 3563 . . . 4 ((𝐴 No 𝑋𝐴) → 𝑋 No )
2 nofun 31046 . . . . 5 (𝑋 No → Fun 𝑋)
3 funrel 5821 . . . . 5 (Fun 𝑋 → Rel 𝑋)
42, 3syl 17 . . . 4 (𝑋 No → Rel 𝑋)
51, 4syl 17 . . 3 ((𝐴 No 𝑋𝐴) → Rel 𝑋)
653adant3 1074 . 2 ((𝐴 No 𝑋𝐴𝐴𝑆) → Rel 𝑋)
7 bdayval 31045 . . . . . 6 (𝑋 No → ( bday 𝑋) = dom 𝑋)
81, 7syl 17 . . . . 5 ((𝐴 No 𝑋𝐴) → ( bday 𝑋) = dom 𝑋)
9 bdaydm 31077 . . . . . . . . 9 dom bday = No
101, 9syl6eleqr 2699 . . . . . . . 8 ((𝐴 No 𝑋𝐴) → 𝑋 ∈ dom bday )
11 bdayfun 31075 . . . . . . . 8 Fun bday
1210, 11jctil 558 . . . . . . 7 ((𝐴 No 𝑋𝐴) → (Fun bday 𝑋 ∈ dom bday ))
13 simpr 476 . . . . . . 7 ((𝐴 No 𝑋𝐴) → 𝑋𝐴)
14 funfvima 6396 . . . . . . 7 ((Fun bday 𝑋 ∈ dom bday ) → (𝑋𝐴 → ( bday 𝑋) ∈ ( bday 𝐴)))
1512, 13, 14sylc 63 . . . . . 6 ((𝐴 No 𝑋𝐴) → ( bday 𝑋) ∈ ( bday 𝐴))
16 elssuni 4403 . . . . . 6 (( bday 𝑋) ∈ ( bday 𝐴) → ( bday 𝑋) ⊆ ( bday 𝐴))
1715, 16syl 17 . . . . 5 ((𝐴 No 𝑋𝐴) → ( bday 𝑋) ⊆ ( bday 𝐴))
188, 17eqsstr3d 3603 . . . 4 ((𝐴 No 𝑋𝐴) → dom 𝑋 ( bday 𝐴))
19183adant3 1074 . . 3 ((𝐴 No 𝑋𝐴𝐴𝑆) → dom 𝑋 ( bday 𝐴))
20 imass2 5420 . . . . 5 (𝐴𝑆 → ( bday 𝐴) ⊆ ( bday 𝑆))
2120unissd 4398 . . . 4 (𝐴𝑆 ( bday 𝐴) ⊆ ( bday 𝑆))
22213ad2ant3 1077 . . 3 ((𝐴 No 𝑋𝐴𝐴𝑆) → ( bday 𝐴) ⊆ ( bday 𝑆))
2319, 22sstrd 3578 . 2 ((𝐴 No 𝑋𝐴𝐴𝑆) → dom 𝑋 ( bday 𝑆))
24 relssres 5357 . 2 ((Rel 𝑋 ∧ dom 𝑋 ( bday 𝑆)) → (𝑋 ( bday 𝑆)) = 𝑋)
256, 23, 24syl2anc 691 1 ((𝐴 No 𝑋𝐴𝐴𝑆) → (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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