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Theorem fcobij 30047
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 2824 . 2 (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6377 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑆𝑇)
42, 3syl 17 . . . . 5 (𝜑𝐺:𝑆𝑇)
54adantr 474 . . . 4 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → 𝐺:𝑆𝑇)
6 fcobij.3 . . . . . 6 (𝜑𝑆𝑉)
7 fcobij.2 . . . . . 6 (𝜑𝑅𝑈)
86, 7elmapd 8135 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 470 . . . 4 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6294 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 581 . . 3 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 8135 . . . 4 (𝜑 → ((𝐺𝑓) ∈ (𝑇𝑚 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1413adantr 474 . . 3 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → ((𝐺𝑓) ∈ (𝑇𝑚 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1511, 14mpbird 249 . 2 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → (𝐺𝑓) ∈ (𝑇𝑚 𝑅))
16 f1ocnv 6389 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6377 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 474 . . . 4 ((𝜑 ∈ (𝑇𝑚 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 8135 . . . . 5 (𝜑 → ( ∈ (𝑇𝑚 𝑅) ↔ :𝑅𝑇))
2120biimpa 470 . . . 4 ((𝜑 ∈ (𝑇𝑚 𝑅)) → :𝑅𝑇)
22 fco 6294 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 581 . . 3 ((𝜑 ∈ (𝑇𝑚 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 8135 . . . 4 (𝜑 → ((𝐺) ∈ (𝑆𝑚 𝑅) ↔ (𝐺):𝑅𝑆))
2524adantr 474 . . 3 ((𝜑 ∈ (𝑇𝑚 𝑅)) → ((𝐺) ∈ (𝑆𝑚 𝑅) ↔ (𝐺):𝑅𝑆))
2623, 25mpbird 249 . 2 ((𝜑 ∈ (𝑇𝑚 𝑅)) → (𝐺) ∈ (𝑆𝑚 𝑅))
27 simpr 479 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → 𝑓 = (𝐺))
2827coeq2d 5516 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 5894 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29syl6eqr 2878 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = ((𝐺𝐺) ∘ ))
31 simpll 785 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → 𝜑)
32 f1ococnv2 6403 . . . . . 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 5515 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → ((𝐺𝐺) ∘ ) = (( I ↾ 𝑇) ∘ ))
35 simplrr 798 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → ∈ (𝑇𝑚 𝑅))
3631, 35, 21syl2anc 581 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6315 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2865 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 479 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5516 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 5894 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42syl6eqr 2878 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = ((𝐺𝐺) ∘ 𝑓))
44 simpll 785 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝜑)
45 f1ococnv1 6405 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5515 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → ((𝐺𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48 simplrl 797 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 ∈ (𝑆𝑚 𝑅))
4944, 48, 9syl2anc 581 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6315 . . . . 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 2865 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 = (𝐺))
5339, 52impbida 837 . 2 ((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) → (𝑓 = (𝐺) ↔ = (𝐺𝑓)))
541, 15, 26, 53f1o2d 7146 1 (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)):(𝑆𝑚 𝑅)–1-1-onto→(𝑇𝑚 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  cmpt 4951   I cid 5248  ccnv 5340  cres 5343  ccom 5345  wf 6118  1-1-ontowf1o 6121  (class class class)co 6904  𝑚 cmap 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-map 8123
This theorem is referenced by: (None)
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