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Theorem fmptco1f1o 29562
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 480 . . . . 5 ((𝜑𝑓𝐴) → 𝑅𝑋)
5 fmptco1f1o.d . . . . . 6 (𝜑𝐷𝑉)
65adantr 480 . . . . 5 ((𝜑𝑓𝐴) → 𝐷𝑉)
7 simpr 476 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐴)
8 fmptco1f1o.a . . . . . . . 8 𝐴 = (𝑅𝑚 𝐸)
97, 8syl6eleq 2740 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓 ∈ (𝑅𝑚 𝐸))
10 elmapi 7921 . . . . . . 7 (𝑓 ∈ (𝑅𝑚 𝐸) → 𝑓:𝐸𝑅)
119, 10syl 17 . . . . . 6 ((𝜑𝑓𝐴) → 𝑓:𝐸𝑅)
12 fmptco1f1o.t . . . . . . . 8 (𝜑𝑇:𝐷1-1-onto𝐸)
13 f1of 6175 . . . . . . . 8 (𝑇:𝐷1-1-onto𝐸𝑇:𝐷𝐸)
1412, 13syl 17 . . . . . . 7 (𝜑𝑇:𝐷𝐸)
1514adantr 480 . . . . . 6 ((𝜑𝑓𝐴) → 𝑇:𝐷𝐸)
16 fco 6096 . . . . . 6 ((𝑓:𝐸𝑅𝑇:𝐷𝐸) → (𝑓𝑇):𝐷𝑅)
1711, 15, 16syl2anc 694 . . . . 5 ((𝜑𝑓𝐴) → (𝑓𝑇):𝐷𝑅)
18 elmapg 7912 . . . . . 6 ((𝑅𝑋𝐷𝑉) → ((𝑓𝑇) ∈ (𝑅𝑚 𝐷) ↔ (𝑓𝑇):𝐷𝑅))
1918biimpar 501 . . . . 5 (((𝑅𝑋𝐷𝑉) ∧ (𝑓𝑇):𝐷𝑅) → (𝑓𝑇) ∈ (𝑅𝑚 𝐷))
204, 6, 17, 19syl21anc 1365 . . . 4 ((𝜑𝑓𝐴) → (𝑓𝑇) ∈ (𝑅𝑚 𝐷))
21 fmptco1f1o.b . . . 4 𝐵 = (𝑅𝑚 𝐷)
2220, 21syl6eleqr 2741 . . 3 ((𝜑𝑓𝐴) → (𝑓𝑇) ∈ 𝐵)
233adantr 480 . . . . 5 ((𝜑𝑔𝐵) → 𝑅𝑋)
24 fmptco1f1o.e . . . . . 6 (𝜑𝐸𝑊)
2524adantr 480 . . . . 5 ((𝜑𝑔𝐵) → 𝐸𝑊)
26 simpr 476 . . . . . . . 8 ((𝜑𝑔𝐵) → 𝑔𝐵)
2726, 21syl6eleq 2740 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑔 ∈ (𝑅𝑚 𝐷))
28 elmapi 7921 . . . . . . 7 (𝑔 ∈ (𝑅𝑚 𝐷) → 𝑔:𝐷𝑅)
2927, 28syl 17 . . . . . 6 ((𝜑𝑔𝐵) → 𝑔:𝐷𝑅)
30 f1ocnv 6187 . . . . . . . 8 (𝑇:𝐷1-1-onto𝐸𝑇:𝐸1-1-onto𝐷)
31 f1of 6175 . . . . . . . 8 (𝑇:𝐸1-1-onto𝐷𝑇:𝐸𝐷)
3212, 30, 313syl 18 . . . . . . 7 (𝜑𝑇:𝐸𝐷)
3332adantr 480 . . . . . 6 ((𝜑𝑔𝐵) → 𝑇:𝐸𝐷)
34 fco 6096 . . . . . 6 ((𝑔:𝐷𝑅𝑇:𝐸𝐷) → (𝑔𝑇):𝐸𝑅)
3529, 33, 34syl2anc 694 . . . . 5 ((𝜑𝑔𝐵) → (𝑔𝑇):𝐸𝑅)
36 elmapg 7912 . . . . . 6 ((𝑅𝑋𝐸𝑊) → ((𝑔𝑇) ∈ (𝑅𝑚 𝐸) ↔ (𝑔𝑇):𝐸𝑅))
3736biimpar 501 . . . . 5 (((𝑅𝑋𝐸𝑊) ∧ (𝑔𝑇):𝐸𝑅) → (𝑔𝑇) ∈ (𝑅𝑚 𝐸))
3823, 25, 35, 37syl21anc 1365 . . . 4 ((𝜑𝑔𝐵) → (𝑔𝑇) ∈ (𝑅𝑚 𝐸))
3938, 8syl6eleqr 2741 . . 3 ((𝜑𝑔𝐵) → (𝑔𝑇) ∈ 𝐴)
40 coass 5692 . . . . . . 7 ((𝑔𝑇) ∘ 𝑇) = (𝑔 ∘ (𝑇𝑇))
4112ad2antrr 762 . . . . . . . . 9 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑇:𝐷1-1-onto𝐸)
42 f1ococnv1 6203 . . . . . . . . . 10 (𝑇:𝐷1-1-onto𝐸 → (𝑇𝑇) = ( I ↾ 𝐷))
4342coeq2d 5317 . . . . . . . . 9 (𝑇:𝐷1-1-onto𝐸 → (𝑔 ∘ (𝑇𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
4441, 43syl 17 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ (𝑇𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
4529adantlr 751 . . . . . . . . 9 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑔:𝐷𝑅)
46 fcoi1 6116 . . . . . . . . 9 (𝑔:𝐷𝑅 → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
4745, 46syl 17 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
4844, 47eqtrd 2685 . . . . . . 7 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 ∘ (𝑇𝑇)) = 𝑔)
4940, 48syl5req 2698 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑔 = ((𝑔𝑇) ∘ 𝑇))
5049eqeq1d 2653 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔 = (𝑓𝑇) ↔ ((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇)))
51 eqcom 2658 . . . . . 6 (((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇) ↔ (𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇))
5251a1i 11 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (((𝑔𝑇) ∘ 𝑇) = (𝑓𝑇) ↔ (𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇)))
53 f1ofo 6182 . . . . . . 7 (𝑇:𝐷1-1-onto𝐸𝑇:𝐷onto𝐸)
5441, 53syl 17 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑇:𝐷onto𝐸)
55 simplr 807 . . . . . . . 8 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓𝐴)
5655, 8syl6eleq 2740 . . . . . . 7 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓 ∈ (𝑅𝑚 𝐸))
57 elmapfn 7922 . . . . . . 7 (𝑓 ∈ (𝑅𝑚 𝐸) → 𝑓 Fn 𝐸)
5856, 57syl 17 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → 𝑓 Fn 𝐸)
59 elmapfn 7922 . . . . . . . 8 ((𝑔𝑇) ∈ (𝑅𝑚 𝐸) → (𝑔𝑇) Fn 𝐸)
6038, 59syl 17 . . . . . . 7 ((𝜑𝑔𝐵) → (𝑔𝑇) Fn 𝐸)
6160adantlr 751 . . . . . 6 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑔𝑇) Fn 𝐸)
62 cocan2 6587 . . . . . 6 ((𝑇:𝐷onto𝐸𝑓 Fn 𝐸 ∧ (𝑔𝑇) Fn 𝐸) → ((𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔𝑇)))
6354, 58, 61, 62syl3anc 1366 . . . . 5 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → ((𝑓𝑇) = ((𝑔𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔𝑇)))
6450, 52, 633bitrrd 295 . . . 4 (((𝜑𝑓𝐴) ∧ 𝑔𝐵) → (𝑓 = (𝑔𝑇) ↔ 𝑔 = (𝑓𝑇)))
6564anasss 680 . . 3 ((𝜑 ∧ (𝑓𝐴𝑔𝐵)) → (𝑓 = (𝑔𝑇) ↔ 𝑔 = (𝑓𝑇)))
662, 22, 39, 65f1o3d 29559 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑔𝐵 ↦ (𝑔𝑇))))
6766simpld 474 1 (𝜑𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  cmpt 4762   I cid 5052  ccnv 5142  cres 5145  ccom 5147   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  (class class class)co 6690  𝑚 cmap 7899
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by:  reprpmtf1o  30832
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