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Theorem fcobij 32419
Description: Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
Hypotheses
Ref Expression
fcobij.1 (𝜑𝐺:𝑆1-1-onto𝑇)
fcobij.2 (𝜑𝑅𝑈)
fcobij.3 (𝜑𝑆𝑉)
fcobij.4 (𝜑𝑇𝑊)
Assertion
Ref Expression
fcobij (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)):(𝑆m 𝑅)–1-1-onto→(𝑇m 𝑅))
Distinct variable groups:   𝑓,𝐺   𝑅,𝑓   𝑆,𝑓   𝑇,𝑓   𝜑,𝑓
Allowed substitution hints:   𝑈(𝑓)   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem fcobij
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . 2 (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6824 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑆𝑇)
42, 3syl 17 . . . . 5 (𝜑𝐺:𝑆𝑇)
54adantr 480 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝐺:𝑆𝑇)
6 fcobij.3 . . . . . 6 (𝜑𝑆𝑉)
7 fcobij.2 . . . . . 6 (𝜑𝑅𝑈)
86, 7elmapd 8831 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 476 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6732 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 583 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 8831 . . . 4 (𝜑 → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1413adantr 480 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1511, 14mpbird 257 . 2 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓) ∈ (𝑇m 𝑅))
16 f1ocnv 6836 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6824 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 480 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 8831 . . . . 5 (𝜑 → ( ∈ (𝑇m 𝑅) ↔ :𝑅𝑇))
2120biimpa 476 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → :𝑅𝑇)
22 fco 6732 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 583 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 8831 . . . 4 (𝜑 → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2524adantr 480 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2623, 25mpbird 257 . 2 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺) ∈ (𝑆m 𝑅))
27 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝑓 = (𝐺))
2827coeq2d 5853 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 6255 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29eqtr4di 2782 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = ((𝐺𝐺) ∘ ))
31 simpll 764 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝜑)
32 f1ococnv2 6851 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑇))
3331, 2, 323syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝐺) = ( I ↾ 𝑇))
3433coeq1d 5852 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ((𝐺𝐺) ∘ ) = (( I ↾ 𝑇) ∘ ))
35 simplrr 775 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ∈ (𝑇m 𝑅))
3631, 35, 21syl2anc 583 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6757 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2769 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5853 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 6255 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42eqtr4di 2782 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = ((𝐺𝐺) ∘ 𝑓))
44 simpll 764 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝜑)
45 f1ococnv1 6853 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5852 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → ((𝐺𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48 simplrl 774 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 ∈ (𝑆m 𝑅))
4944, 48, 9syl2anc 583 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6757 . . . . 5 (𝑓:𝑅𝑆 → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
5149, 50syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
5243, 47, 513eqtrrd 2769 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 = (𝐺))
5339, 52impbida 798 . 2 ((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) → (𝑓 = (𝐺) ↔ = (𝐺𝑓)))
541, 15, 26, 53f1o2d 7654 1 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)):(𝑆m 𝑅)–1-1-onto→(𝑇m 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cmpt 5222   I cid 5564  ccnv 5666  cres 5669  ccom 5671  wf 6530  1-1-ontowf1o 6533  (class class class)co 7402  m cmap 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819
This theorem is referenced by: (None)
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