Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege131d Structured version   Visualization version   GIF version

Theorem frege131d 42128
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 42358. (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 14902 . . . . 5 (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3 imass1 6057 . . . . 5 (𝐹 ⊆ (t+‘𝐹) → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5 ssun2 4137 . . . . 5 ((t+‘𝐹) “ 𝑈) ⊆ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6 ssun2 4137 . . . . 5 (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
75, 6sstri 3957 . . . 4 ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
84, 7sstrdi 3960 . . 3 (𝜑 → (𝐹𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9 trclfvdecomr 42092 . . . . . . . . . . . 12 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
1110cnveqd 5835 . . . . . . . . . 10 (𝜑(t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12 cnvun 6099 . . . . . . . . . . 11 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹((t+‘𝐹) ∘ 𝐹))
13 cnvco 5845 . . . . . . . . . . . 12 ((t+‘𝐹) ∘ 𝐹) = (𝐹(t+‘𝐹))
1413uneq2i 4124 . . . . . . . . . . 11 (𝐹((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1512, 14eqtri 2761 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1611, 15eqtrdi 2789 . . . . . . . . 9 (𝜑(t+‘𝐹) = (𝐹 ∪ (𝐹(t+‘𝐹))))
1716coeq2d 5822 . . . . . . . 8 (𝜑 → (𝐹(t+‘𝐹)) = (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))))
18 coundi 6203 . . . . . . . . 9 (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹))))
19 frege131d.fun . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
20 funcocnv2 6813 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . . 10 (𝜑 → (𝐹𝐹) = ( I ↾ ran 𝐹))
22 coass 6221 . . . . . . . . . . . 12 ((𝐹𝐹) ∘ (t+‘𝐹)) = (𝐹 ∘ (𝐹(t+‘𝐹)))
2322eqcomi 2742 . . . . . . . . . . 11 (𝐹 ∘ (𝐹(t+‘𝐹))) = ((𝐹𝐹) ∘ (t+‘𝐹))
2421coeq1d 5821 . . . . . . . . . . 11 (𝜑 → ((𝐹𝐹) ∘ (t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2523, 24eqtrid 2785 . . . . . . . . . 10 (𝜑 → (𝐹 ∘ (𝐹(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2621, 25uneq12d 4128 . . . . . . . . 9 (𝜑 → ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2718, 26eqtrid 2785 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2817, 27eqtrd 2773 . . . . . . 7 (𝜑 → (𝐹(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2928imaeq1d 6016 . . . . . 6 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈))
30 imaundir 6107 . . . . . 6 ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈))
3129, 30eqtrdi 2789 . . . . 5 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)))
32 resss 5966 . . . . . . . . 9 ( I ↾ ran 𝐹) ⊆ I
33 imass1 6057 . . . . . . . . 9 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
3432, 33ax-mp 5 . . . . . . . 8 (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35 imai 6030 . . . . . . . 8 ( I “ 𝑈) = 𝑈
3634, 35sseqtri 3984 . . . . . . 7 (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37 imaco 6207 . . . . . . . 8 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈))
38 imass1 6057 . . . . . . . . . 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 6030 . . . . . . . . 9 ( I “ ((t+‘𝐹) “ 𝑈)) = ((t+‘𝐹) “ 𝑈)
4139, 40sseqtri 3984 . . . . . . . 8 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ((t+‘𝐹) “ 𝑈)
4237, 41eqsstri 3982 . . . . . . 7 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)
43 unss12 4146 . . . . . . 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 691 . . . . . 6 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝐹) “ 𝑈))
45 ssun1 4136 . . . . . . 7 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46 unass 4130 . . . . . . 7 ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4745, 46sseqtri 3984 . . . . . 6 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4844, 47sstri 3957 . . . . 5 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4931, 48eqsstrdi 4002 . . . 4 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50 coss1 5815 . . . . . . . 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 14910 . . . . . . 7 ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
5351, 52sstrdi 3960 . . . . . 6 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54 imass1 6057 . . . . . 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 3960 . . . 4 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
5749, 56unssd 4150 . . 3 (𝜑 → (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
588, 57unssd 4150 . 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 6017 . . 3 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61 imaundi 6106 . . . 4 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62 imaundi 6106 . . . . . 6 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63 imaco 6207 . . . . . . . 8 ((𝐹(t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6463eqcomi 2742 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹(t+‘𝐹)) “ 𝑈)
65 imaco 6207 . . . . . . . 8 ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6665eqcomi 2742 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
6764, 66uneq12i 4125 . . . . . 6 ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6862, 67eqtri 2761 . . . . 5 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6968uneq2i 4124 . . . 4 ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7061, 69eqtri 2761 . . 3 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7160, 70eqtrdi 2789 . 2 (𝜑 → (𝐹𝐴) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
7258, 71, 593sstr4d 3995 1 (𝜑 → (𝐹𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3447  cun 3912  wss 3914   I cid 5534  ccnv 5636  ran crn 5638  cres 5639  cima 5640  ccom 5641  Fun wfun 6494  cfv 6500  t+ctcl 14879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-seq 13916  df-trcl 14881  df-relexp 14914
This theorem is referenced by:  frege133d  42129
  Copyright terms: Public domain W3C validator