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Theorem frege131d 38373
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 38605. (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 13793 . . . . 5 (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3 imass1 5535 . . . . 5 (𝐹 ⊆ (t+‘𝐹) → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5 ssun2 3810 . . . . 5 ((t+‘𝐹) “ 𝑈) ⊆ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6 ssun2 3810 . . . . 5 (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
75, 6sstri 3645 . . . 4 ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
84, 7syl6ss 3648 . . 3 (𝜑 → (𝐹𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9 trclfvdecomr 38337 . . . . . . . . . . . 12 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
1110cnveqd 5330 . . . . . . . . . 10 (𝜑(t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12 cnvun 5573 . . . . . . . . . . 11 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹((t+‘𝐹) ∘ 𝐹))
13 cnvco 5340 . . . . . . . . . . . 12 ((t+‘𝐹) ∘ 𝐹) = (𝐹(t+‘𝐹))
1413uneq2i 3797 . . . . . . . . . . 11 (𝐹((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1512, 14eqtri 2673 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1611, 15syl6eq 2701 . . . . . . . . 9 (𝜑(t+‘𝐹) = (𝐹 ∪ (𝐹(t+‘𝐹))))
1716coeq2d 5317 . . . . . . . 8 (𝜑 → (𝐹(t+‘𝐹)) = (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))))
18 coundi 5674 . . . . . . . . 9 (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹))))
19 frege131d.fun . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
20 funcocnv2 6199 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . . 10 (𝜑 → (𝐹𝐹) = ( I ↾ ran 𝐹))
22 coass 5692 . . . . . . . . . . . 12 ((𝐹𝐹) ∘ (t+‘𝐹)) = (𝐹 ∘ (𝐹(t+‘𝐹)))
2322eqcomi 2660 . . . . . . . . . . 11 (𝐹 ∘ (𝐹(t+‘𝐹))) = ((𝐹𝐹) ∘ (t+‘𝐹))
2421coeq1d 5316 . . . . . . . . . . 11 (𝜑 → ((𝐹𝐹) ∘ (t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2523, 24syl5eq 2697 . . . . . . . . . 10 (𝜑 → (𝐹 ∘ (𝐹(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2621, 25uneq12d 3801 . . . . . . . . 9 (𝜑 → ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2718, 26syl5eq 2697 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2817, 27eqtrd 2685 . . . . . . 7 (𝜑 → (𝐹(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2928imaeq1d 5500 . . . . . 6 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈))
30 imaundir 5581 . . . . . 6 ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈))
3129, 30syl6eq 2701 . . . . 5 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)))
32 resss 5457 . . . . . . . . 9 ( I ↾ ran 𝐹) ⊆ I
33 imass1 5535 . . . . . . . . 9 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
3432, 33ax-mp 5 . . . . . . . 8 (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35 imai 5513 . . . . . . . 8 ( I “ 𝑈) = 𝑈
3634, 35sseqtri 3670 . . . . . . 7 (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37 imaco 5678 . . . . . . . 8 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈))
38 imass1 5535 . . . . . . . . . 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 5513 . . . . . . . . 9 ( I “ ((t+‘𝐹) “ 𝑈)) = ((t+‘𝐹) “ 𝑈)
4139, 40sseqtri 3670 . . . . . . . 8 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ((t+‘𝐹) “ 𝑈)
4237, 41eqsstri 3668 . . . . . . 7 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)
43 unss12 3818 . . . . . . 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 708 . . . . . 6 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝐹) “ 𝑈))
45 ssun1 3809 . . . . . . 7 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46 unass 3803 . . . . . . 7 ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4745, 46sseqtri 3670 . . . . . 6 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4844, 47sstri 3645 . . . . 5 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4931, 48syl6eqss 3688 . . . 4 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50 coss1 5310 . . . . . . . 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 13801 . . . . . . 7 ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
5351, 52syl6ss 3648 . . . . . 6 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54 imass1 5535 . . . . . 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, 7syl6ss 3648 . . . 4 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
5749, 56unssd 3822 . . 3 (𝜑 → (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
588, 57unssd 3822 . 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 5501 . . 3 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61 imaundi 5580 . . . 4 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62 imaundi 5580 . . . . . 6 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63 imaco 5678 . . . . . . . 8 ((𝐹(t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6463eqcomi 2660 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹(t+‘𝐹)) “ 𝑈)
65 imaco 5678 . . . . . . . 8 ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6665eqcomi 2660 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
6764, 66uneq12i 3798 . . . . . 6 ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6862, 67eqtri 2673 . . . . 5 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6968uneq2i 3797 . . . 4 ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7061, 69eqtri 2673 . . 3 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7160, 70syl6eq 2701 . 2 (𝜑 → (𝐹𝐴) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
7258, 71, 593sstr4d 3681 1 (𝜑 → (𝐹𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607   I cid 5052  ccnv 5142  ran crn 5144  cres 5145  cima 5146  ccom 5147  Fun wfun 5920  cfv 5926  t+ctcl 13770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-trcl 13772  df-relexp 13805
This theorem is referenced by:  frege133d  38374
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