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Theorem fmptco1f1o 29819
Description: The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.)
Hypotheses
Ref Expression
fmptco1f1o.a 𝐴 = (𝑅𝑚 𝐸)
fmptco1f1o.b 𝐵 = (𝑅𝑚 𝐷)
fmptco1f1o.f 𝐹 = (𝑓𝐴 ↦ (𝑓𝑇))
fmptco1f1o.d (𝜑𝐷𝑉)
fmptco1f1o.e (𝜑𝐸𝑊)
fmptco1f1o.r (𝜑𝑅𝑋)
fmptco1f1o.t (𝜑𝑇:𝐷1-1-onto𝐸)
Assertion
Ref Expression
fmptco1f1o (𝜑𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑇,𝑓   𝜑,𝑓
Allowed substitution hints:   𝐷(𝑓)   𝑅(𝑓)   𝐸(𝑓)   𝐹(𝑓)   𝑉(𝑓)   𝑊(𝑓)   𝑋(𝑓)

Proof of Theorem fmptco1f1o
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fmptco1f1o.f . . . 4 𝐹 = (𝑓𝐴 ↦ (𝑓𝑇))
21a1i 11 . . 3 (𝜑𝐹 = (𝑓𝐴 ↦ (𝑓𝑇)))
3 fmptco1f1o.r . . . . . 6 (𝜑𝑅𝑋)
43adantr 472 . . . . 5 ((𝜑𝑓𝐴) → 𝑅𝑋)
5 fmptco1f1o.d . . . . . 6 (𝜑𝐷𝑉)
65adantr 472 . . . . 5 ((𝜑𝑓𝐴) → 𝐷𝑉)
7 simpr 477 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐴)
8 fmptco1f1o.a . . . . . . . 8 𝐴 = (𝑅𝑚 𝐸)
97, 8syl6eleq 2854 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝑅𝑚 𝐸))
10 elmapi 8082 . . . . . . 7 (𝑓 ∈ (𝑅𝑚 𝐸) → 𝑓:𝐸𝑅)
119, 10syl 17 . . . . . 6 ((𝜑𝑓𝐴) → 𝑓:𝐸𝑅)
12 fmptco1f1o.t . . . . . . . 8 (𝜑𝑇:𝐷1-1-onto𝐸)
13 f1of 6320 . . . . . . . 8 (𝑇:𝐷1-1-onto𝐸𝑇:𝐷𝐸)
1412, 13syl 17 . . . . . . 7 (𝜑𝑇:𝐷𝐸)
1514adantr 472 . . . . . 6 ((𝜑𝑓𝐴) → 𝑇:𝐷𝐸)
16 fco 6240 . . . . . 6 ((𝑓:𝐸𝑅𝑇:𝐷𝐸) → (𝑓𝑇):𝐷𝑅)
1711, 15, 16syl2anc 579 . . . . 5 ((𝜑𝑓𝐴) → (𝑓𝑇):𝐷𝑅)
18 elmapg 8073 . . . . . 6 ((𝑅𝑋𝐷𝑉) → ((𝑓𝑇) ∈ (𝑅𝑚 𝐷) ↔ (𝑓𝑇):𝐷𝑅))
1918biimpar 469 . . . . 5 (((𝑅𝑋𝐷𝑉) ∧ (𝑓𝑇):𝐷𝑅) → (𝑓𝑇) ∈ (𝑅𝑚 𝐷))
204, 6, 17, 19syl21anc 866 . . . 4 ((𝜑𝑓𝐴) → (𝑓𝑇) ∈ (𝑅𝑚 𝐷))
21 fmptco1f1o.b . . . 4 𝐵 = (𝑅𝑚 𝐷)
2220, 21syl6eleqr 2855 . . 3 ((𝜑𝑓𝐴) → (𝑓𝑇) ∈ 𝐵)
233adantr 472 . . . . 5 ((𝜑𝑔𝐵) → 𝑅𝑋)
24 fmptco1f1o.e . . . . . 6 (𝜑𝐸𝑊)
2524adantr 472 . . . . 5 ((𝜑𝑔𝐵) → 𝐸𝑊)
26 simpr 477 . . . . . . . 8 ((𝜑𝑔𝐵) → 𝑔𝐵)
2726, 21syl6eleq 2854 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑔 ∈ (𝑅𝑚 𝐷))
28 elmapi 8082 . . . . . . 7 (𝑔 ∈ (𝑅𝑚 𝐷) → 𝑔:𝐷𝑅)
2927, 28syl 17 . . . . . 6 ((𝜑𝑔𝐵) → 𝑔:𝐷𝑅)
30 f1ocnv 6332 . . . . . . . 8 (𝑇:𝐷1-1-onto𝐸𝑇:𝐸1-1-onto𝐷)
31 f1of 6320 . . . . . . . 8 (𝑇:𝐸1-1-onto𝐷𝑇:𝐸𝐷)
3212, 30, 313syl 18 . . . . . . 7 (𝜑𝑇:𝐸𝐷)
3332adantr 472 . . . . . 6 ((𝜑𝑔𝐵) → 𝑇:𝐸𝐷)
34 fco 6240 . . . . . 6 ((𝑔:𝐷𝑅𝑇:𝐸𝐷) → (𝑔𝑇):𝐸𝑅)
3529, 33, 34syl2anc 579 . . . . 5 ((𝜑𝑔𝐵) → (𝑔𝑇):𝐸𝑅)
36 elmapg 8073 . . . . . 6 ((𝑅𝑋𝐸𝑊) → ((𝑔𝑇) ∈ (𝑅𝑚 𝐸) ↔ (𝑔𝑇):𝐸𝑅))
3736biimpar 469 . . . . 5 (((𝑅𝑋𝐸𝑊) ∧ (𝑔𝑇):𝐸𝑅) → (𝑔𝑇) ∈ (𝑅𝑚 𝐸))
3823, 25, 35, 37syl21anc 866 . . . 4 ((𝜑𝑔𝐵) → (𝑔𝑇) ∈ (𝑅𝑚 𝐸))
3938, 8syl6eleqr 2855 . . 3 ((𝜑𝑔𝐵) → (𝑔𝑇) ∈ 𝐴)
40 coass 5840 . . . . . . 7 ((𝑔𝑇) ∘ 𝑇) = (𝑔 ∘ (𝑇𝑇))
4112ad2antrr 717 . . . . . . . . 9 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑇:𝐷1-1-onto𝐸)
42 f1ococnv1 6348 . . . . . . . . . 10 (𝑇:𝐷1-1-onto𝐸 → (𝑇𝑇) = ( I ↾ 𝐷))
4342coeq2d 5453 . . . . . . . . 9 (𝑇:𝐷1-1-onto𝐸 → (𝑔 ∘ (𝑇𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
4441, 43syl 17 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ (𝑇𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
4529adantlr 706 . . . . . . . . 9 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑔:𝐷𝑅)
46 fcoi1 6260 . . . . . . . . 9 (𝑔:𝐷𝑅 → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
4745, 46syl 17 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
4844, 47eqtrd 2799 . . . . . . 7 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ (𝑇𝑇)) = 𝑔)
4940, 48syl5req 2812 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑔 = ((𝑔𝑇) ∘ 𝑇))
5049eqeq1d 2767 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 = (𝑓𝑇) ↔ ((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇)))
51 eqcom 2772 . . . . . 6 (((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇) ↔ (𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇))
5251a1i 11 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇) ↔ (𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇)))
53 f1ofo 6327 . . . . . . 7 (𝑇:𝐷1-1-onto𝐸𝑇:𝐷onto𝐸)
5441, 53syl 17 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑇:𝐷onto𝐸)
55 simplr 785 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓𝐴)
5655, 8syl6eleq 2854 . . . . . . 7 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓 ∈ (𝑅𝑚 𝐸))
57 elmapfn 8083 . . . . . . 7 (𝑓 ∈ (𝑅𝑚 𝐸) → 𝑓 Fn 𝐸)
5856, 57syl 17 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓 Fn 𝐸)
59 elmapfn 8083 . . . . . . . 8 ((𝑔𝑇) ∈ (𝑅𝑚 𝐸) → (𝑔𝑇) Fn 𝐸)
6038, 59syl 17 . . . . . . 7 ((𝜑𝑔𝐵) → (𝑔𝑇) Fn 𝐸)
6160adantlr 706 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔𝑇) Fn 𝐸)
62 cocan2 6739 . . . . . 6 ((𝑇:𝐷onto𝐸𝑓 Fn 𝐸 ∧ (𝑔𝑇) Fn 𝐸) → ((𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔𝑇)))
6354, 58, 61, 62syl3anc 1490 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → ((𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔𝑇)))
6450, 52, 633bitrrd 297 . . . 4 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑓 = (𝑔𝑇) ↔ 𝑔 = (𝑓𝑇)))
6564anasss 458 . . 3 ((𝜑 ∧ (𝑓𝐴𝑔𝐵)) → (𝑓 = (𝑔𝑇) ↔ 𝑔 = (𝑓𝑇)))
662, 22, 39, 65f1o3d 29816 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑔𝐵 ↦ (𝑔𝑇))))
6766simpld 488 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  cmpt 4888   I cid 5184  ccnv 5276  cres 5279  ccom 5281   Fn wfn 6063  wf 6064  ontowfo 6066  1-1-ontowf1o 6067  (class class class)co 6842  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by:  reprpmtf1o  31087
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