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Theorem frege131d 43753
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 43983. (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 15043 . . . . 5 (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3 imass1 6121 . . . . 5 (𝐹 ⊆ (t+‘𝐹) → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5 ssun2 4188 . . . . 5 ((t+‘𝐹) “ 𝑈) ⊆ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6 ssun2 4188 . . . . 5 (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
75, 6sstri 4004 . . . 4 ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
84, 7sstrdi 4007 . . 3 (𝜑 → (𝐹𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9 trclfvdecomr 43717 . . . . . . . . . . . 12 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
1110cnveqd 5888 . . . . . . . . . 10 (𝜑(t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12 cnvun 6164 . . . . . . . . . . 11 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹((t+‘𝐹) ∘ 𝐹))
13 cnvco 5898 . . . . . . . . . . . 12 ((t+‘𝐹) ∘ 𝐹) = (𝐹(t+‘𝐹))
1413uneq2i 4174 . . . . . . . . . . 11 (𝐹((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1512, 14eqtri 2762 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1611, 15eqtrdi 2790 . . . . . . . . 9 (𝜑(t+‘𝐹) = (𝐹 ∪ (𝐹(t+‘𝐹))))
1716coeq2d 5875 . . . . . . . 8 (𝜑 → (𝐹(t+‘𝐹)) = (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))))
18 coundi 6268 . . . . . . . . 9 (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹))))
19 frege131d.fun . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
20 funcocnv2 6873 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . . 10 (𝜑 → (𝐹𝐹) = ( I ↾ ran 𝐹))
22 coass 6286 . . . . . . . . . . . 12 ((𝐹𝐹) ∘ (t+‘𝐹)) = (𝐹 ∘ (𝐹(t+‘𝐹)))
2322eqcomi 2743 . . . . . . . . . . 11 (𝐹 ∘ (𝐹(t+‘𝐹))) = ((𝐹𝐹) ∘ (t+‘𝐹))
2421coeq1d 5874 . . . . . . . . . . 11 (𝜑 → ((𝐹𝐹) ∘ (t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2523, 24eqtrid 2786 . . . . . . . . . 10 (𝜑 → (𝐹 ∘ (𝐹(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2621, 25uneq12d 4178 . . . . . . . . 9 (𝜑 → ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2718, 26eqtrid 2786 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2817, 27eqtrd 2774 . . . . . . 7 (𝜑 → (𝐹(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2928imaeq1d 6078 . . . . . 6 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈))
30 imaundir 6172 . . . . . 6 ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈))
3129, 30eqtrdi 2790 . . . . 5 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)))
32 resss 6021 . . . . . . . . 9 ( I ↾ ran 𝐹) ⊆ I
33 imass1 6121 . . . . . . . . 9 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
3432, 33ax-mp 5 . . . . . . . 8 (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35 imai 6093 . . . . . . . 8 ( I “ 𝑈) = 𝑈
3634, 35sseqtri 4031 . . . . . . 7 (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37 imaco 6272 . . . . . . . 8 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈))
38 imass1 6121 . . . . . . . . . 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 6093 . . . . . . . . 9 ( I “ ((t+‘𝐹) “ 𝑈)) = ((t+‘𝐹) “ 𝑈)
4139, 40sseqtri 4031 . . . . . . . 8 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ((t+‘𝐹) “ 𝑈)
4237, 41eqsstri 4029 . . . . . . 7 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)
43 unss12 4197 . . . . . . 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 692 . . . . . 6 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝐹) “ 𝑈))
45 ssun1 4187 . . . . . . 7 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46 unass 4181 . . . . . . 7 ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4745, 46sseqtri 4031 . . . . . 6 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4844, 47sstri 4004 . . . . 5 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4931, 48eqsstrdi 4049 . . . 4 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50 coss1 5868 . . . . . . . 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 15051 . . . . . . 7 ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
5351, 52sstrdi 4007 . . . . . 6 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54 imass1 6121 . . . . . 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 4007 . . . 4 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
5749, 56unssd 4201 . . 3 (𝜑 → (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
588, 57unssd 4201 . 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 6079 . . 3 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61 imaundi 6171 . . . 4 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62 imaundi 6171 . . . . . 6 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63 imaco 6272 . . . . . . . 8 ((𝐹(t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6463eqcomi 2743 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹(t+‘𝐹)) “ 𝑈)
65 imaco 6272 . . . . . . . 8 ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6665eqcomi 2743 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
6764, 66uneq12i 4175 . . . . . 6 ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6862, 67eqtri 2762 . . . . 5 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6968uneq2i 4174 . . . 4 ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7061, 69eqtri 2762 . . 3 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7160, 70eqtrdi 2790 . 2 (𝜑 → (𝐹𝐴) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
7258, 71, 593sstr4d 4042 1 (𝜑 → (𝐹𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  wss 3962   I cid 5581  ccnv 5687  ran crn 5689  cres 5690  cima 5691  ccom 5692  Fun wfun 6556  cfv 6562  t+ctcl 15020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-seq 14039  df-trcl 15022  df-relexp 15055
This theorem is referenced by:  frege133d  43754
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