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Theorem tfrlemibfn 5883
Description: The union of B is a function defined on x. 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
tfrlemibfn (φ B Fn x)
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 tfrlemibfn
StepHypRef Expression
1 tfrlemisucfn.1 . . . . . 6 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
2 tfrlemisucfn.2 . . . . . 6 (φx(Fun 𝐹 (𝐹x) V))
3 tfrlemi1.3 . . . . . 6 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
4 tfrlemi1.4 . . . . . 6 (φx On)
5 tfrlemi1.5 . . . . . 6 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
61, 2, 3, 4, 5tfrlemibacc 5881 . . . . 5 (φBA)
76unissd 3595 . . . 4 (φ B A)
81recsfval 5872 . . . 4 recs(𝐹) = A
97, 8syl6sseqr 2986 . . 3 (φ B ⊆ recs(𝐹))
101tfrlem7 5874 . . 3 Fun recs(𝐹)
11 funss 4863 . . 3 ( B ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun B))
129, 10, 11mpisyl 1332 . 2 (φ → Fun B)
13 simpr3 911 . . . . . . . . . . . 12 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → = (g ∪ {⟨z, (𝐹g)⟩}))
142ad2antrr 457 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x(Fun 𝐹 (𝐹x) V))
154ad2antrr 457 . . . . . . . . . . . . . . . . . 18 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x On)
16 simplr 482 . . . . . . . . . . . . . . . . . 18 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z x)
17 onelon 4087 . . . . . . . . . . . . . . . . . 18 ((x On z x) → z On)
1815, 16, 17syl2anc 391 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z On)
19 simpr1 909 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g Fn z)
20 simpr2 910 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g A)
211, 14, 18, 19, 20tfrlemisucfn 5879 . . . . . . . . . . . . . . . 16 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z)
22 dffn2 4990 . . . . . . . . . . . . . . . 16 ((g ∪ {⟨z, (𝐹g)⟩}) Fn suc z ↔ (g ∪ {⟨z, (𝐹g)⟩}):suc z⟶V)
2321, 22sylib 127 . . . . . . . . . . . . . . 15 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}):suc z⟶V)
24 fssxp 5001 . . . . . . . . . . . . . . 15 ((g ∪ {⟨z, (𝐹g)⟩}):suc z⟶V → (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (suc z × V))
2523, 24syl 14 . . . . . . . . . . . . . 14 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (suc z × V))
26 eloni 4078 . . . . . . . . . . . . . . . . 17 (x On → Ord x)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → Ord x)
28 ordsucss 4196 . . . . . . . . . . . . . . . 16 (Ord x → (z x → suc zx))
2927, 16, 28sylc 56 . . . . . . . . . . . . . . 15 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → suc zx)
30 xpss1 4391 . . . . . . . . . . . . . . 15 (suc zx → (suc z × V) ⊆ (x × V))
3129, 30syl 14 . . . . . . . . . . . . . 14 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (suc z × V) ⊆ (x × V))
3225, 31sstrd 2949 . . . . . . . . . . . . 13 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (x × V))
33 vex 2554 . . . . . . . . . . . . . . . 16 g V
34 vex 2554 . . . . . . . . . . . . . . . . . 18 z V
352tfrlem3-2d 5869 . . . . . . . . . . . . . . . . . . 19 (φ → (Fun 𝐹 (𝐹g) V))
3635simprd 107 . . . . . . . . . . . . . . . . . 18 (φ → (𝐹g) V)
37 opexg 3955 . . . . . . . . . . . . . . . . . 18 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
3834, 36, 37sylancr 393 . . . . . . . . . . . . . . . . 17 (φ → ⟨z, (𝐹g)⟩ V)
39 snexg 3927 . . . . . . . . . . . . . . . . 17 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (φ → {⟨z, (𝐹g)⟩} V)
41 unexg 4144 . . . . . . . . . . . . . . . 16 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
4233, 40, 41sylancr 393 . . . . . . . . . . . . . . 15 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
43 elpwg 3359 . . . . . . . . . . . . . . 15 ((g ∪ {⟨z, (𝐹g)⟩}) V → ((g ∪ {⟨z, (𝐹g)⟩}) 𝒫 (x × V) ↔ (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (x × V)))
4442, 43syl 14 . . . . . . . . . . . . . 14 (φ → ((g ∪ {⟨z, (𝐹g)⟩}) 𝒫 (x × V) ↔ (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (x × V)))
4544ad2antrr 457 . . . . . . . . . . . . 13 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → ((g ∪ {⟨z, (𝐹g)⟩}) 𝒫 (x × V) ↔ (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (x × V)))
4632, 45mpbird 156 . . . . . . . . . . . 12 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) 𝒫 (x × V))
4713, 46eqeltrd 2111 . . . . . . . . . . 11 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → 𝒫 (x × V))
4847ex 108 . . . . . . . . . 10 ((φ z x) → ((g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → 𝒫 (x × V)))
4948exlimdv 1697 . . . . . . . . 9 ((φ z x) → (g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → 𝒫 (x × V)))
5049rexlimdva 2427 . . . . . . . 8 (φ → (z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → 𝒫 (x × V)))
5150abssdv 3008 . . . . . . 7 (φ → {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))} ⊆ 𝒫 (x × V))
523, 51syl5eqss 2983 . . . . . 6 (φB ⊆ 𝒫 (x × V))
53 sspwuni 3730 . . . . . 6 (B ⊆ 𝒫 (x × V) ↔ B ⊆ (x × V))
5452, 53sylib 127 . . . . 5 (φ B ⊆ (x × V))
55 dmss 4477 . . . . 5 ( B ⊆ (x × V) → dom B ⊆ dom (x × V))
5654, 55syl 14 . . . 4 (φ → dom B ⊆ dom (x × V))
57 dmxpss 4696 . . . 4 dom (x × V) ⊆ x
5856, 57syl6ss 2951 . . 3 (φ → dom Bx)
591, 2, 3, 4, 5tfrlemibxssdm 5882 . . 3 (φx ⊆ dom B)
6058, 59eqssd 2956 . 2 (φ → dom B = x)
61 df-fn 4848 . 2 ( B Fn x ↔ (Fun B dom B = x))
6212, 60, 61sylanbrc 394 1 (φ B Fn x)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909  wss 2911  𝒫 cpw 3351  {csn 3367  cop 3370   cuni 3571  Ord word 4065  Oncon0 4066  suc csuc 4068   × cxp 4286  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  wf 4841  cfv 4845  recscrecs 5860
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-in1 544  ax-in2 545  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  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfrlemibex  5884  tfrlemiubacc  5885  tfrlemiex  5886
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