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Theorem fcobij 30371
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 2826 . 2 (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)) = (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓))
2 fcobij.1 . . . . . 6 (𝜑𝐺:𝑆1-1-onto𝑇)
3 f1of 6612 . . . . . 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 8410 . . . . 5 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↔ 𝑓:𝑅𝑆))
98biimpa 477 . . . 4 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → 𝑓:𝑅𝑆)
10 fco 6528 . . . 4 ((𝐺:𝑆𝑇𝑓:𝑅𝑆) → (𝐺𝑓):𝑅𝑇)
115, 9, 10syl2anc 584 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓):𝑅𝑇)
12 fcobij.4 . . . . 5 (𝜑𝑇𝑊)
1312, 7elmapd 8410 . . . 4 (𝜑 → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1413adantr 481 . . 3 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → ((𝐺𝑓) ∈ (𝑇m 𝑅) ↔ (𝐺𝑓):𝑅𝑇))
1511, 14mpbird 258 . 2 ((𝜑𝑓 ∈ (𝑆m 𝑅)) → (𝐺𝑓) ∈ (𝑇m 𝑅))
16 f1ocnv 6624 . . . . . 6 (𝐺:𝑆1-1-onto𝑇𝐺:𝑇1-1-onto𝑆)
17 f1of 6612 . . . . . 6 (𝐺:𝑇1-1-onto𝑆𝐺:𝑇𝑆)
182, 16, 173syl 18 . . . . 5 (𝜑𝐺:𝑇𝑆)
1918adantr 481 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → 𝐺:𝑇𝑆)
2012, 7elmapd 8410 . . . . 5 (𝜑 → ( ∈ (𝑇m 𝑅) ↔ :𝑅𝑇))
2120biimpa 477 . . . 4 ((𝜑 ∈ (𝑇m 𝑅)) → :𝑅𝑇)
22 fco 6528 . . . 4 ((𝐺:𝑇𝑆:𝑅𝑇) → (𝐺):𝑅𝑆)
2319, 21, 22syl2anc 584 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺):𝑅𝑆)
246, 7elmapd 8410 . . . 4 (𝜑 → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2524adantr 481 . . 3 ((𝜑 ∈ (𝑇m 𝑅)) → ((𝐺) ∈ (𝑆m 𝑅) ↔ (𝐺):𝑅𝑆))
2623, 25mpbird 258 . 2 ((𝜑 ∈ (𝑇m 𝑅)) → (𝐺) ∈ (𝑆m 𝑅))
27 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝑓 = (𝐺))
2827coeq2d 5732 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = (𝐺 ∘ (𝐺)))
29 coass 6116 . . . . 5 ((𝐺𝐺) ∘ ) = (𝐺 ∘ (𝐺))
3028, 29syl6eqr 2879 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (𝐺𝑓) = ((𝐺𝐺) ∘ ))
31 simpll 763 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → 𝜑)
32 f1ococnv2 6638 . . . . . 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 5731 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ((𝐺𝐺) ∘ ) = (( I ↾ 𝑇) ∘ ))
35 simplrr 774 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → ∈ (𝑇m 𝑅))
3631, 35, 21syl2anc 584 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → :𝑅𝑇)
37 fcoi2 6550 . . . . 5 (:𝑅𝑇 → (( I ↾ 𝑇) ∘ ) = )
3836, 37syl 17 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → (( I ↾ 𝑇) ∘ ) = )
3930, 34, 383eqtrrd 2866 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ 𝑓 = (𝐺)) → = (𝐺𝑓))
40 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → = (𝐺𝑓))
4140coeq2d 5732 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = (𝐺 ∘ (𝐺𝑓)))
42 coass 6116 . . . . 5 ((𝐺𝐺) ∘ 𝑓) = (𝐺 ∘ (𝐺𝑓))
4341, 42syl6eqr 2879 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺) = ((𝐺𝐺) ∘ 𝑓))
44 simpll 763 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝜑)
45 f1ococnv1 6640 . . . . . 6 (𝐺:𝑆1-1-onto𝑇 → (𝐺𝐺) = ( I ↾ 𝑆))
4644, 2, 453syl 18 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → (𝐺𝐺) = ( I ↾ 𝑆))
4746coeq1d 5731 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → ((𝐺𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48 simplrl 773 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 ∈ (𝑆m 𝑅))
4944, 48, 9syl2anc 584 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓:𝑅𝑆)
50 fcoi2 6550 . . . . 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 2866 . . 3 (((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) ∧ = (𝐺𝑓)) → 𝑓 = (𝐺))
5339, 52impbida 797 . 2 ((𝜑 ∧ (𝑓 ∈ (𝑆m 𝑅) ∧ ∈ (𝑇m 𝑅))) → (𝑓 = (𝐺) ↔ = (𝐺𝑓)))
541, 15, 26, 53f1o2d 7389 1 (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)):(𝑆m 𝑅)–1-1-onto→(𝑇m 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  cmpt 5143   I cid 5458  ccnv 5553  cres 5556  ccom 5558  wf 6348  1-1-ontowf1o 6351  (class class class)co 7148  m cmap 8396
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398
This theorem is referenced by: (None)
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