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Theorem fcobij 29474
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 2620 . 2 (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6124 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑆𝑇)
42, 3syl 17 . . . . 5 (𝜑𝐺:𝑆𝑇)
54adantr 481 . . . 4 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → 𝐺:𝑆𝑇)
6 fcobij.3 . . . . . 6 (𝜑𝑆𝑉)
7 fcobij.2 . . . . . 6 (𝜑𝑅𝑈)
86, 7elmapd 7856 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 501 . . . 4 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6045 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 692 . . 3 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 7856 . . . 4 (𝜑 → ((𝐺𝑓) ∈ (𝑇𝑚 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1413adantr 481 . . 3 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → ((𝐺𝑓) ∈ (𝑇𝑚 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1511, 14mpbird 247 . 2 ((𝜑𝑓 ∈ (𝑆𝑚 𝑅)) → (𝐺𝑓) ∈ (𝑇𝑚 𝑅))
16 f1ocnv 6136 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6124 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 481 . . . 4 ((𝜑 ∈ (𝑇𝑚 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 7856 . . . . 5 (𝜑 → ( ∈ (𝑇𝑚 𝑅) ↔ :𝑅𝑇))
2120biimpa 501 . . . 4 ((𝜑 ∈ (𝑇𝑚 𝑅)) → :𝑅𝑇)
22 fco 6045 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 692 . . 3 ((𝜑 ∈ (𝑇𝑚 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 7856 . . . 4 (𝜑 → ((𝐺) ∈ (𝑆𝑚 𝑅) ↔ (𝐺):𝑅𝑆))
2524adantr 481 . . 3 ((𝜑 ∈ (𝑇𝑚 𝑅)) → ((𝐺) ∈ (𝑆𝑚 𝑅) ↔ (𝐺):𝑅𝑆))
2623, 25mpbird 247 . 2 ((𝜑 ∈ (𝑇𝑚 𝑅)) → (𝐺) ∈ (𝑆𝑚 𝑅))
27 simpr 477 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → 𝑓 = (𝐺))
2827coeq2d 5273 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 5642 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29syl6eqr 2672 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = ((𝐺𝐺) ∘ ))
31 simpll 789 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → 𝜑)
32 f1ococnv2 6150 . . . . . 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 5272 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → ((𝐺𝐺) ∘ ) = (( I ↾ 𝑇) ∘ ))
35 simplrr 800 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → ∈ (𝑇𝑚 𝑅))
3631, 35, 21syl2anc 692 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6066 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2659 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 477 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5273 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 5642 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42syl6eqr 2672 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = ((𝐺𝐺) ∘ 𝑓))
44 simpll 789 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝜑)
45 f1ococnv1 6152 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5272 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → ((𝐺𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48 simplrl 799 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 ∈ (𝑆𝑚 𝑅))
4944, 48, 9syl2anc 692 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6066 . . . . 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 2659 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 = (𝐺))
5339, 52impbida 876 . 2 ((𝜑 ∧ (𝑓 ∈ (𝑆𝑚 𝑅) ∧ ∈ (𝑇𝑚 𝑅))) → (𝑓 = (𝐺) ↔ = (𝐺𝑓)))
541, 15, 26, 53f1o2d 6872 1 (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)):(𝑆𝑚 𝑅)–1-1-onto→(𝑇𝑚 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  cmpt 4720   I cid 5013  ccnv 5103  cres 5106  ccom 5108  wf 5872  1-1-ontowf1o 5875  (class class class)co 6635  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844
This theorem is referenced by: (None)
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