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Theorem 2fvcoidd 7291
Description: Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
2fvcoidd.f (𝜑𝐹:𝐴𝐵)
2fvcoidd.g (𝜑𝐺:𝐵𝐴)
2fvcoidd.i (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
Assertion
Ref Expression
2fvcoidd (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
Distinct variable groups:   𝐴,𝑎   𝐹,𝑎   𝐺,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)

Proof of Theorem 2fvcoidd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2fvcoidd.g . . 3 (𝜑𝐺:𝐵𝐴)
2 2fvcoidd.f . . 3 (𝜑𝐹:𝐴𝐵)
3 fcompt 7127 . . 3 ((𝐺:𝐵𝐴𝐹:𝐴𝐵) → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
41, 2, 3syl2anc 583 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
5 2fvcoidd.i . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
6 2fveq3 6890 . . . . . . . 8 (𝑎 = 𝑥 → (𝐺‘(𝐹𝑎)) = (𝐺‘(𝐹𝑥)))
7 id 22 . . . . . . . 8 (𝑎 = 𝑥𝑎 = 𝑥)
86, 7eqeq12d 2742 . . . . . . 7 (𝑎 = 𝑥 → ((𝐺‘(𝐹𝑎)) = 𝑎 ↔ (𝐺‘(𝐹𝑥)) = 𝑥))
98rspccv 3603 . . . . . 6 (∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
105, 9syl 17 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
1110imp 406 . . . 4 ((𝜑𝑥𝐴) → (𝐺‘(𝐹𝑥)) = 𝑥)
1211mpteq2dva 5241 . . 3 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = (𝑥𝐴𝑥))
13 mptresid 6044 . . 3 ( I ↾ 𝐴) = (𝑥𝐴𝑥)
1412, 13eqtr4di 2784 . 2 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = ( I ↾ 𝐴))
154, 14eqtrd 2766 1 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3055  cmpt 5224   I cid 5566  cres 5671  ccom 5673  wf 6533  cfv 6537
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by:  2fvidf1od  7292  2fvidinvd  7293
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