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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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