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Theorem tfrlemibxssdm 5882
Description: The union of B is defined on all ordinals. Lemma for tfrlemi1 5887. (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 5869 . . . . . . . . . . 11 (φ → (Fun 𝐹 (𝐹g) V))
54simprd 107 . . . . . . . . . 10 (φ → (𝐹g) V)
653ad2ant1 924 . . . . . . . . 9 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → (𝐹g) V)
7 vex 2554 . . . . . . . . . . . . 13 z V
8 opexg 3955 . . . . . . . . . . . . 13 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
97, 5, 8sylancr 393 . . . . . . . . . . . 12 (φ → ⟨z, (𝐹g)⟩ V)
10 snidg 3392 . . . . . . . . . . . 12 (⟨z, (𝐹g)⟩ V → ⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩})
11 elun2 3105 . . . . . . . . . . . 12 (⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩} → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
129, 10, 113syl 17 . . . . . . . . . . 11 (φ → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
13123ad2ant1 924 . . . . . . . . . 10 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
14 simp2r 930 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → z x)
15 simp3l 931 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → g Fn z)
16 onelon 4087 . . . . . . . . . . . . . . 15 ((x On z x) → z On)
17 rspe 2364 . . . . . . . . . . . . . . 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 2554 . . . . . . . . . . . . . . 15 g V
2119, 20tfrlem3a 5866 . . . . . . . . . . . . . 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 921 . . . . . . . . . . . 12 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → g A)
2414, 15, 233jca 1083 . . . . . . . . . . 11 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → (z x g Fn z g A))
25 snexg 3927 . . . . . . . . . . . . . 14 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
26 unexg 4144 . . . . . . . . . . . . . . 15 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
2720, 26mpan 400 . . . . . . . . . . . . . 14 ({⟨z, (𝐹g)⟩} V → (g ∪ {⟨z, (𝐹g)⟩}) V)
289, 25, 273syl 17 . . . . . . . . . . . . 13 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
29 isset 2555 . . . . . . . . . . . . 13 ((g ∪ {⟨z, (𝐹g)⟩}) V ↔ = (g ∪ {⟨z, (𝐹g)⟩}))
3028, 29sylib 127 . . . . . . . . . . . 12 (φ = (g ∪ {⟨z, (𝐹g)⟩}))
31303ad2ant1 924 . . . . . . . . . . 11 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → = (g ∪ {⟨z, (𝐹g)⟩}))
32 simpr3 911 . . . . . . . . . . . . . . 15 ((z x (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → = (g ∪ {⟨z, (𝐹g)⟩}))
33 19.8a 1479 . . . . . . . . . . . . . . . 16 ((g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})))
34 rspe 2364 . . . . . . . . . . . . . . . . 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 2145 . . . . . . . . . . . . . . . . 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 2112 . . . . . . . . . . . . . 14 ((z x (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) B)
40393exp2 1121 . . . . . . . . . . . . 13 (z x → (g Fn z → (g A → ( = (g ∪ {⟨z, (𝐹g)⟩}) → (g ∪ {⟨z, (𝐹g)⟩}) B))))
41403imp 1097 . . . . . . . . . . . 12 ((z x g Fn z g A) → ( = (g ∪ {⟨z, (𝐹g)⟩}) → (g ∪ {⟨z, (𝐹g)⟩}) B))
4241exlimdv 1697 . . . . . . . . . . 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 3576 . . . . . . . . . 10 ((⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) (g ∪ {⟨z, (𝐹g)⟩}) B) → ⟨z, (𝐹g)⟩ B)
4513, 43, 44syl2anc 391 . . . . . . . . 9 ((φ (x On z x) (g Fn z w z (gw) = (𝐹‘(gw)))) → ⟨z, (𝐹g)⟩ B)
46 opeq2 3541 . . . . . . . . . . . 12 (w = (𝐹g) → ⟨z, w⟩ = ⟨z, (𝐹g)⟩)
4746eleq1d 2103 . . . . . . . . . . 11 (w = (𝐹g) → (⟨z, w B ↔ ⟨z, (𝐹g)⟩ B))
4847spcegv 2635 . . . . . . . . . 10 ((𝐹g) V → (⟨z, (𝐹g)⟩ Bwz, w B))
497eldm2 4476 . . . . . . . . . 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 1105 . . . . . . 7 ((φ (x On z x)) → ((g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5352exlimdv 1697 . . . . . 6 ((φ (x On z x)) → (g(g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5453anassrs 380 . . . . 5 (((φ x On) z x) → (g(g Fn z w z (gw) = (𝐹‘(gw))) → z dom B))
5554ralimdva 2381 . . . 4 ((φ x On) → (z x g(g Fn z w z (gw) = (𝐹‘(gw))) → z x z dom B))
562, 55mpdan 398 . . 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 2929 . 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 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909  wss 2911  {csn 3367  cop 3370   cuni 3571  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlemibfn  5883
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