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Theorem fcobij 32498
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 2728 . 2 (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6833 . . . . . 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 8852 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 476 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6741 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 583 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 8852 . . . 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 6845 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6833 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 480 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 8852 . . . . 5 (𝜑 → ( ∈ (𝑇m 𝑅) ↔ :𝑅𝑇))
2120biimpa 476 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → :𝑅𝑇)
22 fco 6741 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 583 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 8852 . . . 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 5859 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 6263 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29eqtr4di 2786 . . . 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 6860 . . . . . 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 5858 . . . 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 583 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6766 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2773 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5859 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 6263 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42eqtr4di 2786 . . . 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 6862 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5858 . . . 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 583 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6766 . . . . 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 2773 . . 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 7669 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 1534  wcel 2099  cmpt 5225   I cid 5569  ccnv 5671  cres 5674  ccom 5676  wf 6538  1-1-ontowf1o 6541  (class class class)co 7414  m cmap 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840
This theorem is referenced by: (None)
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