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Theorem tfrlemibxssdm 5837
Description: The union of B is defined on all ordinals. Lemma for tfrlemi1 5842. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemisucfn.2 (φx(Fun 𝐹 (𝐹x) V))
tfrlemi1.3 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
tfrlemi1.4 (φx On)
tfrlemi1.5 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
Assertion
Ref Expression
tfrlemibxssdm (φx ⊆ dom B)
Distinct variable groups:   f,g,,w,x,y,z,A   f,𝐹,g,,w,x,y,z   φ,w,y   w,B,f,g,,z   φ,g,,z
Allowed substitution hints:   φ(x,f)   B(x,y)

Proof of Theorem tfrlemibxssdm
StepHypRef Expression
1 tfrlemi1.5 . . 3 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
2 tfrlemi1.4 . . . 4 (φx On)
3 tfrlemisucfn.2 . . . . . . . . . . . 12 (φx(Fun 𝐹 (𝐹x) V))
43tfrlem3-2d 5825 . . . . . . . . . . 11 (φ → (Fun 𝐹 (𝐹g) V))
54simprd 107 . . . . . . . . . 10 (φ → (𝐹g) V)
653ad2ant1 916 . . . . . . . . 9 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → (𝐹g) V)
7 vex 2537 . . . . . . . . . . . . 13 z V
8 opexg 3917 . . . . . . . . . . . . 13 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
97, 5, 8sylancr 395 . . . . . . . . . . . 12 (φ → ⟨z, (𝐹g)⟩ V)
10 snidg 3353 . . . . . . . . . . . 12 (⟨z, (𝐹g)⟩ V → ⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩})
11 elun2 3090 . . . . . . . . . . . 12 (⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩} → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
129, 10, 113syl 17 . . . . . . . . . . 11 (φ → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
13123ad2ant1 916 . . . . . . . . . 10 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
14 simp2r 922 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → z x)
15 simp3l 923 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → g Fn z)
16 onelon 4046 . . . . . . . . . . . . . . 15 ((x On z x) → z On)
17 rspe 2347 . . . . . . . . . . . . . . 15 ((z On (g Fn z w z (gw) = (𝐹‘(gw)))) → z On (g Fn z w z (gw) = (𝐹‘(gw))))
1816, 17sylan 267 . . . . . . . . . . . . . 14 (((x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → z On (g Fn z w z (gw) = (𝐹‘(gw))))
19 tfrlemisucfn.1 . . . . . . . . . . . . . . 15 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
20 vex 2537 . . . . . . . . . . . . . . 15 g V
2119, 20tfrlem3a 5822 . . . . . . . . . . . . . 14 (g Az On (g Fn z w z (gw) = (𝐹‘(gw))))
2218, 21sylibr 137 . . . . . . . . . . . . 13 (((x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → g A)
23223adant1 913 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → g A)
2414, 15, 233jca 1072 . . . . . . . . . . 11 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → (z x g Fn z g A))
25 snexg 3889 . . . . . . . . . . . . . 14 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
26 unexg 4104 . . . . . . . . . . . . . . 15 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
2720, 26mpan 402 . . . . . . . . . . . . . 14 ({⟨z, (𝐹g)⟩} V → (g ∪ {⟨z, (𝐹g)⟩}) V)
289, 25, 273syl 17 . . . . . . . . . . . . 13 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
29 isset 2538 . . . . . . . . . . . . 13 ((g ∪ {⟨z, (𝐹g)⟩}) V ↔ = (g ∪ {⟨z, (𝐹g)⟩}))
3028, 29sylib 127 . . . . . . . . . . . 12 (φ = (g ∪ {⟨z, (𝐹g)⟩}))
31303ad2ant1 916 . . . . . . . . . . 11 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → = (g ∪ {⟨z, (𝐹g)⟩}))
32 simpr3 903 . . . . . . . . . . . . . . 15 ((z x (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → = (g ∪ {⟨z, (𝐹g)⟩}))
33 19.8a 1466 . . . . . . . . . . . . . . . 16 ((g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})))
34 rspe 2347 . . . . . . . . . . . . . . . . 17 ((z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})))
35 tfrlemi1.3 . . . . . . . . . . . . . . . . . 18 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
3635abeq2i 2131 . . . . . . . . . . . . . . . . 17 ( Bz x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})))
3734, 36sylibr 137 . . . . . . . . . . . . . . . 16 ((z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → B)
3833, 37sylan2 270 . . . . . . . . . . . . . . 15 ((z x (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → B)
3932, 38eqeltrrd 2098 . . . . . . . . . . . . . 14 ((z x (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) B)
40393exp2 1111 . . . . . . . . . . . . 13 (z x → (g Fn z → (g A → ( = (g ∪ {⟨z, (𝐹g)⟩}) → (g ∪ {⟨z, (𝐹g)⟩}) B))))
41403imp 1087 . . . . . . . . . . . 12 ((z x g Fn z g A) → ( = (g ∪ {⟨z, (𝐹g)⟩}) → (g ∪ {⟨z, (𝐹g)⟩}) B))
4241exlimdv 1683 . . . . . . . . . . 11 ((z x g Fn z g A) → ( = (g ∪ {⟨z, (𝐹g)⟩}) → (g ∪ {⟨z, (𝐹g)⟩}) B))
4324, 31, 42sylc 56 . . . . . . . . . 10 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → (g ∪ {⟨z, (𝐹g)⟩}) B)
44 elunii 3538 . . . . . . . . . 10 ((⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) (g ∪ {⟨z, (𝐹g)⟩}) B) → ⟨z, (𝐹g)⟩ B)
4513, 43, 44syl2anc 393 . . . . . . . . 9 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → ⟨z, (𝐹g)⟩ B)
46 opeq2 3503 . . . . . . . . . . . 12 (w = (𝐹g) → ⟨z, w⟩ = ⟨z, (𝐹g)⟩)
4746eleq1d 2089 . . . . . . . . . . 11 (w = (𝐹g) → (⟨z, w B ↔ ⟨z, (𝐹g)⟩ B))
4847spcegv 2617 . . . . . . . . . 10 ((𝐹g) V → (⟨z, (𝐹g)⟩ Bwz, w B))
497eldm2 4436 . . . . . . . . . 10 (z dom Bwz, w B)
5048, 49syl6ibr 151 . . . . . . . . 9 ((𝐹g) V → (⟨z, (𝐹g)⟩ Bz dom B))
516, 45, 50sylc 56 . . . . . . . 8 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → z dom B)
52513expia 1095 . . . . . . 7 ((φ (x On z x)) → ((g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5352exlimdv 1683 . . . . . 6 ((φ (x On z x)) → (g(g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5453anassrs 382 . . . . 5 (((φ x On) z x) → (g(g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5554ralimdva 2364 . . . 4 ((φ x On) → (z x g(g Fn z w z (gw) = (𝐹‘(gw))) → z x z dom B))
562, 55mpdan 400 . . 3 (φ → (z x g(g Fn z w z (gw) = (𝐹‘(gw))) → z x z dom B))
571, 56mpd 13 . 2 (φz x z dom B)
58 dfss3 2914 . 2 (x ⊆ dom Bz x z dom B)
5957, 58sylibr 137 1 (φx ⊆ dom B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 876  wal 1315  wex 1362   = wceq 1374   wcel 1376  {cab 2009  wral 2283  wrex 2284  Vcvv 2534  cun 2894  wss 2896  {csn 3328  cop 3331   cuni 3533  Oncon0 4026  dom cdm 4248  cres 4250  Fun wfun 4800   Fn wfn 4801  cfv 4806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-13 1387  ax-14 1388  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005  ax-sep 3828  ax-pow 3880  ax-pr 3897  ax-un 4096
This theorem depends on definitions:  df-bi 110  df-3an 878  df-tru 1232  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2901  df-in 2903  df-ss 2910  df-pw 3314  df-sn 3334  df-pr 3335  df-op 3337  df-uni 3534  df-br 3718  df-opab 3772  df-tr 3808  df-iord 4029  df-on 4030  df-xp 4254  df-rel 4255  df-cnv 4256  df-co 4257  df-dm 4258  df-res 4260  df-iota 4771  df-fun 4808  df-fn 4809  df-fv 4814
This theorem is referenced by:  tfrlemibfn  5838
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