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Theorem 2fvcoidd 7294
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 7130 . . 3 ((𝐺:𝐵𝐴𝐹:𝐴𝐵) → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
41, 2, 3syl2anc 584 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
5 2fvcoidd.i . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
6 2fveq3 6896 . . . . . . . 8 (𝑎 = 𝑥 → (𝐺‘(𝐹𝑎)) = (𝐺‘(𝐹𝑥)))
7 id 22 . . . . . . . 8 (𝑎 = 𝑥𝑎 = 𝑥)
86, 7eqeq12d 2748 . . . . . . 7 (𝑎 = 𝑥 → ((𝐺‘(𝐹𝑎)) = 𝑎 ↔ (𝐺‘(𝐹𝑥)) = 𝑥))
98rspccv 3609 . . . . . 6 (∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
105, 9syl 17 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
1110imp 407 . . . 4 ((𝜑𝑥𝐴) → (𝐺‘(𝐹𝑥)) = 𝑥)
1211mpteq2dva 5248 . . 3 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = (𝑥𝐴𝑥))
13 mptresid 6050 . . 3 ( I ↾ 𝐴) = (𝑥𝐴𝑥)
1412, 13eqtr4di 2790 . 2 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = ( I ↾ 𝐴))
154, 14eqtrd 2772 1 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3061  cmpt 5231   I cid 5573  cres 5678  ccom 5680  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  2fvidf1od  7295  2fvidinvd  7296
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