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Theorem frege131d 42505
Description: If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 42735. (Contributed by RP, 17-Jul-2020.)
Hypotheses
Ref Expression
frege131d.f (𝜑𝐹 ∈ V)
frege131d.a (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
frege131d.fun (𝜑 → Fun 𝐹)
Assertion
Ref Expression
frege131d (𝜑 → (𝐹𝐴) ⊆ 𝐴)

Proof of Theorem frege131d
StepHypRef Expression
1 frege131d.f . . . . 5 (𝜑𝐹 ∈ V)
2 trclfvlb 14954 . . . . 5 (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3 imass1 6100 . . . . 5 (𝐹 ⊆ (t+‘𝐹) → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5 ssun2 4173 . . . . 5 ((t+‘𝐹) “ 𝑈) ⊆ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6 ssun2 4173 . . . . 5 (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
75, 6sstri 3991 . . . 4 ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
84, 7sstrdi 3994 . . 3 (𝜑 → (𝐹𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9 trclfvdecomr 42469 . . . . . . . . . . . 12 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
1110cnveqd 5875 . . . . . . . . . 10 (𝜑(t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12 cnvun 6142 . . . . . . . . . . 11 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹((t+‘𝐹) ∘ 𝐹))
13 cnvco 5885 . . . . . . . . . . . 12 ((t+‘𝐹) ∘ 𝐹) = (𝐹(t+‘𝐹))
1413uneq2i 4160 . . . . . . . . . . 11 (𝐹((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1512, 14eqtri 2760 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1611, 15eqtrdi 2788 . . . . . . . . 9 (𝜑(t+‘𝐹) = (𝐹 ∪ (𝐹(t+‘𝐹))))
1716coeq2d 5862 . . . . . . . 8 (𝜑 → (𝐹(t+‘𝐹)) = (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))))
18 coundi 6246 . . . . . . . . 9 (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹))))
19 frege131d.fun . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
20 funcocnv2 6858 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . . 10 (𝜑 → (𝐹𝐹) = ( I ↾ ran 𝐹))
22 coass 6264 . . . . . . . . . . . 12 ((𝐹𝐹) ∘ (t+‘𝐹)) = (𝐹 ∘ (𝐹(t+‘𝐹)))
2322eqcomi 2741 . . . . . . . . . . 11 (𝐹 ∘ (𝐹(t+‘𝐹))) = ((𝐹𝐹) ∘ (t+‘𝐹))
2421coeq1d 5861 . . . . . . . . . . 11 (𝜑 → ((𝐹𝐹) ∘ (t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2523, 24eqtrid 2784 . . . . . . . . . 10 (𝜑 → (𝐹 ∘ (𝐹(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2621, 25uneq12d 4164 . . . . . . . . 9 (𝜑 → ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2718, 26eqtrid 2784 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2817, 27eqtrd 2772 . . . . . . 7 (𝜑 → (𝐹(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2928imaeq1d 6058 . . . . . 6 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈))
30 imaundir 6150 . . . . . 6 ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈))
3129, 30eqtrdi 2788 . . . . 5 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)))
32 resss 6006 . . . . . . . . 9 ( I ↾ ran 𝐹) ⊆ I
33 imass1 6100 . . . . . . . . 9 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
3432, 33ax-mp 5 . . . . . . . 8 (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35 imai 6073 . . . . . . . 8 ( I “ 𝑈) = 𝑈
3634, 35sseqtri 4018 . . . . . . 7 (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37 imaco 6250 . . . . . . . 8 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈))
38 imass1 6100 . . . . . . . . . 10 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ( I “ ((t+‘𝐹) “ 𝑈)))
3932, 38ax-mp 5 . . . . . . . . 9 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ( I “ ((t+‘𝐹) “ 𝑈))
40 imai 6073 . . . . . . . . 9 ( I “ ((t+‘𝐹) “ 𝑈)) = ((t+‘𝐹) “ 𝑈)
4139, 40sseqtri 4018 . . . . . . . 8 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ((t+‘𝐹) “ 𝑈)
4237, 41eqsstri 4016 . . . . . . 7 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)
43 unss12 4182 . . . . . . 7 (((( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈 ∧ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)) → ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝐹) “ 𝑈)))
4436, 42, 43mp2an 690 . . . . . 6 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝐹) “ 𝑈))
45 ssun1 4172 . . . . . . 7 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46 unass 4166 . . . . . . 7 ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4745, 46sseqtri 4018 . . . . . 6 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4844, 47sstri 3991 . . . . 5 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4931, 48eqsstrdi 4036 . . . 4 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50 coss1 5855 . . . . . . . 8 (𝐹 ⊆ (t+‘𝐹) → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
511, 2, 503syl 18 . . . . . . 7 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ ((t+‘𝐹) ∘ (t+‘𝐹)))
52 trclfvcotrg 14962 . . . . . . 7 ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
5351, 52sstrdi 3994 . . . . . 6 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54 imass1 6100 . . . . . 6 ((𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹) → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5553, 54syl 17 . . . . 5 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5655, 7sstrdi 3994 . . . 4 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
5749, 56unssd 4186 . . 3 (𝜑 → (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
588, 57unssd 4186 . 2 (𝜑 → ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
59 frege131d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
6059imaeq2d 6059 . . 3 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61 imaundi 6149 . . . 4 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62 imaundi 6149 . . . . . 6 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63 imaco 6250 . . . . . . . 8 ((𝐹(t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6463eqcomi 2741 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹(t+‘𝐹)) “ 𝑈)
65 imaco 6250 . . . . . . . 8 ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6665eqcomi 2741 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
6764, 66uneq12i 4161 . . . . . 6 ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6862, 67eqtri 2760 . . . . 5 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6968uneq2i 4160 . . . 4 ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7061, 69eqtri 2760 . . 3 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7160, 70eqtrdi 2788 . 2 (𝜑 → (𝐹𝐴) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
7258, 71, 593sstr4d 4029 1 (𝜑 → (𝐹𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  cun 3946  wss 3948   I cid 5573  ccnv 5675  ran crn 5677  cres 5678  cima 5679  ccom 5680  Fun wfun 6537  cfv 6543  t+ctcl 14931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-seq 13966  df-trcl 14933  df-relexp 14966
This theorem is referenced by:  frege133d  42506
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