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Theorem tfrlemibfn 5859
 Description: The union of B is a function defined on x. Lemma for tfrlemi1 5863. (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 5857 . . . . 5 (φBA)
76unissd 3574 . . . 4 (φ B A)
81recsfval 5849 . . . 4 recs(𝐹) = A
97, 8syl6sseqr 2965 . . 3 (φ B ⊆ recs(𝐹))
101tfrlem7 5851 . . 3 Fun recs(𝐹)
11 funss 4842 . . 3 ( B ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun B))
129, 10, 11mpisyl 1311 . 2 (φ → Fun B)
13 simpr3 898 . . . . . . . . . . . 12 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → = (g ∪ {⟨z, (𝐹g)⟩}))
142ad2antrr 460 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x(Fun 𝐹 (𝐹x) V))
154ad2antrr 460 . . . . . . . . . . . . . . . . . 18 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x On)
16 simplr 470 . . . . . . . . . . . . . . . . . 18 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z x)
17 onelon 4066 . . . . . . . . . . . . . . . . . 18 ((x On z x) → z On)
1815, 16, 17syl2anc 393 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z On)
19 simpr1 896 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g Fn z)
20 simpr2 897 . . . . . . . . . . . . . . . . 17 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g A)
211, 14, 18, 19, 20tfrlemisucfn 5855 . . . . . . . . . . . . . . . 16 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z)
22 dffn2 4969 . . . . . . . . . . . . . . . 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 4979 . . . . . . . . . . . . . . 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 4057 . . . . . . . . . . . . . . . . 17 (x On → Ord x)
2715, 26syl 14 . . . . . . . . . . . . . . . 16 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → Ord x)
28 ordsucss 4176 . . . . . . . . . . . . . . . 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 4371 . . . . . . . . . . . . . . 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 2928 . . . . . . . . . . . . 13 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) ⊆ (x × V))
33 vex 2534 . . . . . . . . . . . . . . . 16 g V
34 vex 2534 . . . . . . . . . . . . . . . . . 18 z V
352tfrlem3-2d 5846 . . . . . . . . . . . . . . . . . . 19 (φ → (Fun 𝐹 (𝐹g) V))
3635simprd 107 . . . . . . . . . . . . . . . . . 18 (φ → (𝐹g) V)
37 opexg 3934 . . . . . . . . . . . . . . . . . 18 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
3834, 36, 37sylancr 395 . . . . . . . . . . . . . . . . 17 (φ → ⟨z, (𝐹g)⟩ V)
39 snexg 3906 . . . . . . . . . . . . . . . . 17 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
4038, 39syl 14 . . . . . . . . . . . . . . . 16 (φ → {⟨z, (𝐹g)⟩} V)
41 unexg 4124 . . . . . . . . . . . . . . . 16 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
4233, 40, 41sylancr 395 . . . . . . . . . . . . . . 15 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
43 elpwg 3338 . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 2092 . . . . . . . . . . 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 1678 . . . . . . . . 9 ((φ z x) → (g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → 𝒫 (x × V)))
5049rexlimdva 2407 . . . . . . . 8 (φ → (z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → 𝒫 (x × V)))
5150abssdv 2987 . . . . . . 7 (φ → {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))} ⊆ 𝒫 (x × V))
523, 51syl5eqss 2962 . . . . . 6 (φB ⊆ 𝒫 (x × V))
53 sspwuni 3709 . . . . . 6 (B ⊆ 𝒫 (x × V) ↔ B ⊆ (x × V))
5452, 53sylib 127 . . . . 5 (φ B ⊆ (x × V))
55 dmss 4457 . . . . 5 ( B ⊆ (x × V) → dom B ⊆ dom (x × V))
5654, 55syl 14 . . . 4 (φ → dom B ⊆ dom (x × V))
57 dmxpss 4676 . . . 4 dom (x × V) ⊆ x
5856, 57syl6ss 2930 . . 3 (φ → dom Bx)
591, 2, 3, 4, 5tfrlemibxssdm 5858 . . 3 (φx ⊆ dom B)
6058, 59eqssd 2935 . 2 (φ → dom B = x)
61 df-fn 4828 . 2 ( B Fn x ↔ (Fun B dom B = x))
6212, 60, 61sylanbrc 396 1 (φ B Fn x)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871  ∀wal 1224   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004  ∀wral 2280  ∃wrex 2281  Vcvv 2531   ∪ cun 2888   ⊆ wss 2890  𝒫 cpw 3330  {csn 3346  ⟨cop 3349  ∪ cuni 3550  Ord word 4044  Oncon0 4045  suc csuc 4047   × cxp 4266  dom cdm 4268   ↾ cres 4270  Fun wfun 4819   Fn wfn 4820  ⟶wf 4821  ‘cfv 4825  recscrecs 5837 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-fv 4833  df-recs 5838 This theorem is referenced by:  tfrlemibex  5860  tfrlemiubacc  5861  tfrlemiex  5862
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