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Theorem fcobij 31693
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 2733 . 2 (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6788 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑆𝑇)
42, 3syl 17 . . . . 5 (𝜑𝐺:𝑆𝑇)
54adantr 482 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝐺:𝑆𝑇)
6 fcobij.3 . . . . . 6 (𝜑𝑆𝑉)
7 fcobij.2 . . . . . 6 (𝜑𝑅𝑈)
86, 7elmapd 8785 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 478 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6696 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 585 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 8785 . . . 4 (𝜑 → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1413adantr 482 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1511, 14mpbird 257 . 2 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓) ∈ (𝑇m 𝑅))
16 f1ocnv 6800 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6788 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 482 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 8785 . . . . 5 (𝜑 → ( ∈ (𝑇m 𝑅) ↔ :𝑅𝑇))
2120biimpa 478 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → :𝑅𝑇)
22 fco 6696 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 585 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 8785 . . . 4 (𝜑 → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2524adantr 482 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2623, 25mpbird 257 . 2 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺) ∈ (𝑆m 𝑅))
27 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝑓 = (𝐺))
2827coeq2d 5822 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 6221 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29eqtr4di 2791 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = ((𝐺𝐺) ∘ ))
31 simpll 766 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝜑)
32 f1ococnv2 6815 . . . . . 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 5821 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ((𝐺𝐺) ∘ ) = (( I ↾ 𝑇) ∘ ))
35 simplrr 777 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ∈ (𝑇m 𝑅))
3631, 35, 21syl2anc 585 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6721 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2778 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5822 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 6221 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42eqtr4di 2791 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = ((𝐺𝐺) ∘ 𝑓))
44 simpll 766 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝜑)
45 f1ococnv1 6817 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5821 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → ((𝐺𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48 simplrl 776 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 ∈ (𝑆m 𝑅))
4944, 48, 9syl2anc 585 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6721 . . . . 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 2778 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 = (𝐺))
5339, 52impbida 800 . 2 ((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) → (𝑓 = (𝐺) ↔ = (𝐺𝑓)))
541, 15, 26, 53f1o2d 7611 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 397   = wceq 1542  wcel 2107  cmpt 5192   I cid 5534  ccnv 5636  cres 5639  ccom 5641  wf 6496  1-1-ontowf1o 6499  (class class class)co 7361  m cmap 8771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773
This theorem is referenced by: (None)
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