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Theorem frege124d 37072
 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 37301. (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 2616 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) = 𝐴)
5 frege124d.x . . . . . . . . . . . 12 (𝜑𝑋 ∈ dom 𝐹)
6 funbrfvb 6148 . . . . . . . . . . . 12 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = 𝐴𝑋𝐹𝐴))
72, 5, 6syl2anc 691 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑋) = 𝐴𝑋𝐹𝐴))
84, 7mpbid 221 . . . . . . . . . 10 (𝜑𝑋𝐹𝐴)
9 funeu 5828 . . . . . . . . . 10 ((Fun 𝐹𝑋𝐹𝐴) → ∃!𝑎 𝑋𝐹𝑎)
102, 8, 9syl2anc 691 . . . . . . . . 9 (𝜑 → ∃!𝑎 𝑋𝐹𝑎)
11 fvex 6113 . . . . . . . . . . . . 13 (𝐹𝑋) ∈ V
121, 11syl6eqel 2696 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
13 sbcan 3445 . . . . . . . . . . . . 13 ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) ↔ ([𝐴 / 𝑎]𝑋𝐹𝑎[𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵))
14 sbcbr2g 4640 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑋𝐹𝑎𝑋𝐹𝐴 / 𝑎𝑎))
15 csbvarg 3955 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → 𝐴 / 𝑎𝑎 = 𝐴)
1615breq2d 4595 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (𝑋𝐹𝐴 / 𝑎𝑎𝑋𝐹𝐴))
1714, 16bitrd 267 . . . . . . . . . . . . . 14 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑋𝐹𝑎𝑋𝐹𝐴))
18 sbcng 3443 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → ([𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵 ↔ ¬ [𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵))
19 sbcbr1g 4639 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵𝐴 / 𝑎𝑎(t+‘𝐹)𝐵))
2015breq1d 4593 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ V → (𝐴 / 𝑎𝑎(t+‘𝐹)𝐵𝐴(t+‘𝐹)𝐵))
2119, 20bitrd 267 . . . . . . . . . . . . . . . 16 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵𝐴(t+‘𝐹)𝐵))
2221notbid 307 . . . . . . . . . . . . . . 15 (𝐴 ∈ V → (¬ [𝐴 / 𝑎]𝑎(t+‘𝐹)𝐵 ↔ ¬ 𝐴(t+‘𝐹)𝐵))
2318, 22bitrd 267 . . . . . . . . . . . . . 14 (𝐴 ∈ V → ([𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵 ↔ ¬ 𝐴(t+‘𝐹)𝐵))
2417, 23anbi12d 743 . . . . . . . . . . . . 13 (𝐴 ∈ V → (([𝐴 / 𝑎]𝑋𝐹𝑎[𝐴 / 𝑎] ¬ 𝑎(t+‘𝐹)𝐵) ↔ (𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵)))
2513, 24syl5bb 271 . . . . . . . . . . . 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 3487 . . . . . . . . . . 11 ([𝐴 / 𝑎](𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵) → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵))
2826, 27syl6bir 243 . . . . . . . . . 10 (𝜑 → ((𝑋𝐹𝐴 ∧ ¬ 𝐴(t+‘𝐹)𝐵) → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)))
298, 28mpand 707 . . . . . . . . 9 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)))
30 eupicka 2525 . . . . . . . . 9 ((∃!𝑎 𝑋𝐹𝑎 ∧ ∃𝑎(𝑋𝐹𝑎 ∧ ¬ 𝑎(t+‘𝐹)𝐵)) → ∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵))
3110, 29, 30syl6an 566 . . . . . . . 8 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵)))
32 frege124d.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
33 funrel 5821 . . . . . . . . . . . . . 14 (Fun 𝐹 → Rel 𝐹)
342, 33syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
35 reltrclfv 13606 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ Rel 𝐹) → Rel (t+‘𝐹))
3632, 34, 35syl2anc 691 . . . . . . . . . . . 12 (𝜑 → Rel (t+‘𝐹))
37 brrelex2 5081 . . . . . . . . . . . 12 ((Rel (t+‘𝐹) ∧ 𝑋(t+‘𝐹)𝐵) → 𝐵 ∈ V)
3836, 3, 37syl2anc 691 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
39 brcog 5210 . . . . . . . . . . 11 ((𝑋 ∈ dom 𝐹𝐵 ∈ V) → (𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
405, 38, 39syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
4140notbid 307 . . . . . . . . 9 (𝜑 → (¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵 ↔ ¬ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵)))
42 alinexa 1759 . . . . . . . . 9 (∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵) ↔ ¬ ∃𝑎(𝑋𝐹𝑎𝑎(t+‘𝐹)𝐵))
4341, 42syl6rbbr 278 . . . . . . . 8 (𝜑 → (∀𝑎(𝑋𝐹𝑎 → ¬ 𝑎(t+‘𝐹)𝐵) ↔ ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
4431, 43sylibd 228 . . . . . . 7 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
45 brdif 4635 . . . . . . . 8 (𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵 ↔ (𝑋(t+‘𝐹)𝐵 ∧ ¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵))
4645simplbi2 653 . . . . . . 7 (𝑋(t+‘𝐹)𝐵 → (¬ 𝑋((t+‘𝐹) ∘ 𝐹)𝐵𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵))
473, 44, 46sylsyld 59 . . . . . 6 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵))
48 trclfvdecomr 37039 . . . . . . . . . . 11 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
4932, 48syl 17 . . . . . . . . . 10 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
50 uncom 3719 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹)
5149, 50syl6eq 2660 . . . . . . . . 9 (𝜑 → (t+‘𝐹) = (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹))
52 eqimss 3620 . . . . . . . . 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 4004 . . . . . . . 8 ((t+‘𝐹) ⊆ (((t+‘𝐹) ∘ 𝐹) ∪ 𝐹) ↔ ((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹)) ⊆ 𝐹)
5553, 54sylib 207 . . . . . . 7 (𝜑 → ((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹)) ⊆ 𝐹)
5655ssbrd 4626 . . . . . 6 (𝜑 → (𝑋((t+‘𝐹) ∖ ((t+‘𝐹) ∘ 𝐹))𝐵𝑋𝐹𝐵))
5747, 56syld 46 . . . . 5 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝑋𝐹𝐵))
58 funbrfv 6144 . . . . 5 (Fun 𝐹 → (𝑋𝐹𝐵 → (𝐹𝑋) = 𝐵))
592, 57, 58sylsyld 59 . . . 4 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵 → (𝐹𝑋) = 𝐵))
60 eqcom 2617 . . . 4 ((𝐹𝑋) = 𝐵𝐵 = (𝐹𝑋))
6159, 60syl6ib 240 . . 3 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝐵 = (𝐹𝑋)))
62 eqtr3 2631 . . 3 ((𝐴 = (𝐹𝑋) ∧ 𝐵 = (𝐹𝑋)) → 𝐴 = 𝐵)
631, 61, 62syl6an 566 . 2 (𝜑 → (¬ 𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
6463orrd 392 1 (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804  t+ctcl 13572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-trcl 13574  df-relexp 13609 This theorem is referenced by:  frege126d  37073
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