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Theorem frege124d 37872
Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 38101. (Contributed by RP, 16-Jul-2020.)
Hypotheses
Ref Expression
frege124d.f (𝜑𝐹 ∈ V)
frege124d.x (𝜑𝑋 ∈ dom 𝐹)
frege124d.a (𝜑𝐴 = (𝐹𝑋))
frege124d.xb (𝜑𝑋(t+‘𝐹)𝐵)
frege124d.fun (𝜑 → Fun 𝐹)
Assertion
Ref Expression
frege124d (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Proof of Theorem frege124d
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 frege124d.a . . 3 (𝜑𝐴 = (𝐹𝑋))
2 frege124d.fun . . . . 5 (𝜑 → Fun 𝐹)
3 frege124d.xb . . . . . . 7 (𝜑𝑋(t+‘𝐹)𝐵)
41eqcomd 2626 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) = 𝐴)
5 frege124d.x . . . . . . . . . . . 12 (𝜑𝑋 ∈ dom 𝐹)
6 funbrfvb 6225 . . . . . . . . . . . 12 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = 𝐴𝑋𝐹𝐴))
72, 5, 6syl2anc 692 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑋) = 𝐴𝑋𝐹𝐴))
84, 7mpbid 222 . . . . . . . . . 10 (𝜑𝑋𝐹𝐴)
9 funeu 5901 . . . . . . . . . 10 ((Fun 𝐹𝑋𝐹𝐴) → ∃!𝑎 𝑋𝐹𝑎)
102, 8, 9syl2anc 692 . . . . . . . . 9 (𝜑 → ∃!𝑎 𝑋𝐹𝑎)
11 fvex 6188 . . . . . . . . . . . . 13 (𝐹𝑋) ∈ V
121, 11syl6eqel 2707 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
13 sbcan 3472 . . . . . . . . . . . . 13 ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) ↔ ([𝐴 / 𝑎]𝑋𝐹𝑎[𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵))
14 sbcbr2g 4701 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑋𝐹𝑎𝑋𝐹𝐴 / 𝑎𝑎))
15 csbvarg 3994 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → 𝐴 / 𝑎𝑎 = 𝐴)
1615breq2d 4656 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (𝑋𝐹𝐴 / 𝑎𝑎𝑋𝐹𝐴))
1714, 16bitrd 268 . . . . . . . . . . . . . 14 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑋𝐹𝑎𝑋𝐹𝐴))
18 sbcng 3470 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → ([𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵 ↔ ¬ [𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵))
19 sbcbr1g 4700 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵𝐴 / 𝑎𝑎(t+‘𝐹)𝐵))
2015breq1d 4654 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝐴 / 𝑎𝑎(t+‘𝐹)𝐵𝐴(t+‘𝐹)𝐵))
2119, 20bitrd 268 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵𝐴(t+‘𝐹)𝐵))
2221notbid 308 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (¬ [𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵 ↔ ¬ 𝐴(t+‘𝐹)𝐵))
2318, 22bitrd 268 . . . . . . . . . . . . . 14 (𝐴 ∈ V → ([𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵 ↔ ¬ 𝐴(t+‘𝐹)𝐵))
2417, 23anbi12d 746 . . . . . . . . . . . . 13 (𝐴 ∈ V → (([𝐴 / 𝑎]𝑋𝐹𝑎[𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵) ↔ (𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵)))
2513, 24syl5bb 272 . . . . . . . . . . . 12 (𝐴 ∈ V → ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) ↔ (𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵)))
2612, 25syl 17 . . . . . . . . . . 11 (𝜑 → ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) ↔ (𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵)))
27 spesbc 3514 . . . . . . . . . . 11 ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵))
2826, 27syl6bir 244 . . . . . . . . . 10 (𝜑 → ((𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵) → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)))
298, 28mpand 710 . . . . . . . . 9 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)))
30 eupicka 2535 . . . . . . . . 9 ((∃!𝑎 𝑋𝐹𝑎 ∧ ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)) → ∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵))
3110, 29, 30syl6an 567 . . . . . . . 8 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵)))
32 frege124d.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
33 funrel 5893 . . . . . . . . . . . . . 14 (Fun 𝐹 → Rel 𝐹)
342, 33syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
35 reltrclfv 13739 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ Rel 𝐹) → Rel (t+‘𝐹))
3632, 34, 35syl2anc 692 . . . . . . . . . . . 12 (𝜑 → Rel (t+‘𝐹))
37 brrelex2 5147 . . . . . . . . . . . 12 ((Rel (t+‘𝐹) ∧ 𝑋(t+‘𝐹)𝐵) → 𝐵 ∈ V)
3836, 3, 37syl2anc 692 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
39 brcog 5277 . . . . . . . . . . 11 ((𝑋 ∈ dom 𝐹𝐵 ∈ V) → (𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
405, 38, 39syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
4140notbid 308 . . . . . . . . 9 (𝜑 → (¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ¬ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
42 alinexa 1768 . . . . . . . . 9 (∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵) ↔ ¬ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵))
4341, 42syl6rbbr 279 . . . . . . . 8 (𝜑 → (∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵) ↔ ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
4431, 43sylibd 229 . . . . . . 7 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
45 brdif 4696 . . . . . . . 8 (𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵 ↔ (𝑋(t+‘𝐹)𝐵 ∧ ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
4645simplbi2 654 . . . . . . 7 (𝑋(t+‘𝐹)𝐵 → (¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵))
473, 44, 46sylsyld 61 . . . . . 6 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵))
48 trclfvdecomr 37839 . . . . . . . . . . 11 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
4932, 48syl 17 . . . . . . . . . 10 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
50 uncom 3749 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹)
5149, 50syl6eq 2670 . . . . . . . . 9 (𝜑 → (t+‘𝐹) = (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹))
52 eqimss 3649 . . . . . . . . 9 ((t+‘𝐹) = (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹) → (t+‘𝐹) ⊆ (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹))
5351, 52syl 17 . . . . . . . 8 (𝜑 → (t+‘𝐹) ⊆ (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹))
54 ssundif 4043 . . . . . . . 8 ((t+‘𝐹) ⊆ (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹) ↔ ((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹)) ⊆ 𝐹)
5553, 54sylib 208 . . . . . . 7 (𝜑 → ((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹)) ⊆ 𝐹)
5655ssbrd 4687 . . . . . 6 (𝜑 → (𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵𝑋𝐹𝐵))
5747, 56syld 47 . . . . 5 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝑋𝐹𝐵))
58 funbrfv 6221 . . . . 5 (Fun 𝐹 → (𝑋𝐹𝐵 → (𝐹𝑋) = 𝐵))
592, 57, 58sylsyld 61 . . . 4 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → (𝐹𝑋) = 𝐵))
60 eqcom 2627 . . . 4 ((𝐹𝑋) = 𝐵𝐵 = (𝐹𝑋))
6159, 60syl6ib 241 . . 3 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝐵 = (𝐹𝑋)))
62 eqtr3 2641 . . 3 ((𝐴 = (𝐹𝑋) ∧ 𝐵 = (𝐹𝑋)) → 𝐴 = 𝐵)
631, 61, 62syl6an 567 . 2 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
6463orrd 393 1 (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  ∃!weu 2468  Vcvv 3195  [wsbc 3429  csb 3526  cdif 3564  cun 3565  wss 3567   class class class wbr 4644  dom cdm 5104  ccom 5108  Rel wrel 5109  Fun wfun 5870  cfv 5876  t+ctcl 13705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-seq 12785  df-trcl 13707  df-relexp 13742
This theorem is referenced by:  frege126d  37873
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