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Theorem frege131d 42026
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 42256. (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 14893 . . . . 5 (𝐹 ∈ V → 𝐹 ⊆ (t+‘𝐹))
3 imass1 6053 . . . . 5 (𝐹 ⊆ (t+‘𝐹) → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝐹𝑈) ⊆ ((t+‘𝐹) “ 𝑈))
5 ssun2 4133 . . . . 5 ((t+‘𝐹) “ 𝑈) ⊆ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))
6 ssun2 4133 . . . . 5 (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
75, 6sstri 3953 . . . 4 ((t+‘𝐹) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
84, 7sstrdi 3956 . . 3 (𝜑 → (𝐹𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
9 trclfvdecomr 41990 . . . . . . . . . . . 12 (𝐹 ∈ V → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
101, 9syl 17 . . . . . . . . . . 11 (𝜑 → (t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
1110cnveqd 5831 . . . . . . . . . 10 (𝜑(t+‘𝐹) = (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)))
12 cnvun 6095 . . . . . . . . . . 11 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹((t+‘𝐹) ∘ 𝐹))
13 cnvco 5841 . . . . . . . . . . . 12 ((t+‘𝐹) ∘ 𝐹) = (𝐹(t+‘𝐹))
1413uneq2i 4120 . . . . . . . . . . 11 (𝐹((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1512, 14eqtri 2764 . . . . . . . . . 10 (𝐹 ∪ ((t+‘𝐹) ∘ 𝐹)) = (𝐹 ∪ (𝐹(t+‘𝐹)))
1611, 15eqtrdi 2792 . . . . . . . . 9 (𝜑(t+‘𝐹) = (𝐹 ∪ (𝐹(t+‘𝐹))))
1716coeq2d 5818 . . . . . . . 8 (𝜑 → (𝐹(t+‘𝐹)) = (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))))
18 coundi 6199 . . . . . . . . 9 (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹))))
19 frege131d.fun . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
20 funcocnv2 6809 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . . 10 (𝜑 → (𝐹𝐹) = ( I ↾ ran 𝐹))
22 coass 6217 . . . . . . . . . . . 12 ((𝐹𝐹) ∘ (t+‘𝐹)) = (𝐹 ∘ (𝐹(t+‘𝐹)))
2322eqcomi 2745 . . . . . . . . . . 11 (𝐹 ∘ (𝐹(t+‘𝐹))) = ((𝐹𝐹) ∘ (t+‘𝐹))
2421coeq1d 5817 . . . . . . . . . . 11 (𝜑 → ((𝐹𝐹) ∘ (t+‘𝐹)) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2523, 24eqtrid 2788 . . . . . . . . . 10 (𝜑 → (𝐹 ∘ (𝐹(t+‘𝐹))) = (( I ↾ ran 𝐹) ∘ (t+‘𝐹)))
2621, 25uneq12d 4124 . . . . . . . . 9 (𝜑 → ((𝐹𝐹) ∪ (𝐹 ∘ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2718, 26eqtrid 2788 . . . . . . . 8 (𝜑 → (𝐹 ∘ (𝐹 ∪ (𝐹(t+‘𝐹)))) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2817, 27eqtrd 2776 . . . . . . 7 (𝜑 → (𝐹(t+‘𝐹)) = (( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))))
2928imaeq1d 6012 . . . . . 6 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈))
30 imaundir 6103 . . . . . 6 ((( I ↾ ran 𝐹) ∪ (( I ↾ ran 𝐹) ∘ (t+‘𝐹))) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈))
3129, 30eqtrdi 2792 . . . . 5 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) = ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)))
32 resss 5962 . . . . . . . . 9 ( I ↾ ran 𝐹) ⊆ I
33 imass1 6053 . . . . . . . . 9 (( I ↾ ran 𝐹) ⊆ I → (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈))
3432, 33ax-mp 5 . . . . . . . 8 (( I ↾ ran 𝐹) “ 𝑈) ⊆ ( I “ 𝑈)
35 imai 6026 . . . . . . . 8 ( I “ 𝑈) = 𝑈
3634, 35sseqtri 3980 . . . . . . 7 (( I ↾ ran 𝐹) “ 𝑈) ⊆ 𝑈
37 imaco 6203 . . . . . . . 8 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) = (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈))
38 imass1 6053 . . . . . . . . . 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 6026 . . . . . . . . 9 ( I “ ((t+‘𝐹) “ 𝑈)) = ((t+‘𝐹) “ 𝑈)
4139, 40sseqtri 3980 . . . . . . . 8 (( I ↾ ran 𝐹) “ ((t+‘𝐹) “ 𝑈)) ⊆ ((t+‘𝐹) “ 𝑈)
4237, 41eqsstri 3978 . . . . . . 7 ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈) ⊆ ((t+‘𝐹) “ 𝑈)
43 unss12 4142 . . . . . . 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 4132 . . . . . . 7 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈))
46 unass 4126 . . . . . . 7 ((𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ∪ ((t+‘𝐹) “ 𝑈)) = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4745, 46sseqtri 3980 . . . . . 6 (𝑈 ∪ ((t+‘𝐹) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4844, 47sstri 3953 . . . . 5 ((( I ↾ ran 𝐹) “ 𝑈) ∪ ((( I ↾ ran 𝐹) ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))
4931, 48eqsstrdi 3998 . . . 4 (𝜑 → ((𝐹(t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
50 coss1 5811 . . . . . . . 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 14901 . . . . . . 7 ((t+‘𝐹) ∘ (t+‘𝐹)) ⊆ (t+‘𝐹)
5351, 52sstrdi 3956 . . . . . 6 (𝜑 → (𝐹 ∘ (t+‘𝐹)) ⊆ (t+‘𝐹))
54 imass1 6053 . . . . . 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 3956 . . . 4 (𝜑 → ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
5749, 56unssd 4146 . . 3 (𝜑 → (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)) ⊆ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
588, 57unssd 4146 . 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 6013 . . 3 (𝜑 → (𝐹𝐴) = (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))))
61 imaundi 6102 . . . 4 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))
62 imaundi 6102 . . . . . 6 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈)))
63 imaco 6203 . . . . . . . 8 ((𝐹(t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6463eqcomi 2745 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹(t+‘𝐹)) “ 𝑈)
65 imaco 6203 . . . . . . . 8 ((𝐹 ∘ (t+‘𝐹)) “ 𝑈) = (𝐹 “ ((t+‘𝐹) “ 𝑈))
6665eqcomi 2745 . . . . . . 7 (𝐹 “ ((t+‘𝐹) “ 𝑈)) = ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)
6764, 66uneq12i 4121 . . . . . 6 ((𝐹 “ ((t+‘𝐹) “ 𝑈)) ∪ (𝐹 “ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6862, 67eqtri 2764 . . . . 5 (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))) = (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))
6968uneq2i 4120 . . . 4 ((𝐹𝑈) ∪ (𝐹 “ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7061, 69eqtri 2764 . . 3 (𝐹 “ (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈)))
7160, 70eqtrdi 2792 . 2 (𝜑 → (𝐹𝐴) = ((𝐹𝑈) ∪ (((𝐹(t+‘𝐹)) “ 𝑈) ∪ ((𝐹 ∘ (t+‘𝐹)) “ 𝑈))))
7258, 71, 593sstr4d 3991 1 (𝜑 → (𝐹𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  wss 3910   I cid 5530  ccnv 5632  ran crn 5634  cres 5635  cima 5636  ccom 5637  Fun wfun 6490  cfv 6496  t+ctcl 14870
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-trcl 14872  df-relexp 14905
This theorem is referenced by:  frege133d  42027
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