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Theorem 2fvcoidd 6512
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 6360 . . 3 ((𝐺:𝐵𝐴𝐹:𝐴𝐵) → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
41, 2, 3syl2anc 692 . 2 (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))
5 2fvcoidd.i . . . . . 6 (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)
6 fveq2 6153 . . . . . . . . 9 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
76fveq2d 6157 . . . . . . . 8 (𝑎 = 𝑥 → (𝐺‘(𝐹𝑎)) = (𝐺‘(𝐹𝑥)))
8 id 22 . . . . . . . 8 (𝑎 = 𝑥𝑎 = 𝑥)
97, 8eqeq12d 2636 . . . . . . 7 (𝑎 = 𝑥 → ((𝐺‘(𝐹𝑎)) = 𝑎 ↔ (𝐺‘(𝐹𝑥)) = 𝑥))
109rspccv 3295 . . . . . 6 (∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
115, 10syl 17 . . . . 5 (𝜑 → (𝑥𝐴 → (𝐺‘(𝐹𝑥)) = 𝑥))
1211imp 445 . . . 4 ((𝜑𝑥𝐴) → (𝐺‘(𝐹𝑥)) = 𝑥)
1312mpteq2dva 4709 . . 3 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = (𝑥𝐴𝑥))
14 mptresid 5420 . . 3 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
1513, 14syl6eq 2671 . 2 (𝜑 → (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))) = ( I ↾ 𝐴))
164, 15eqtrd 2655 1 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  cmpt 4678   I cid 4989  cres 5081  ccom 5083  wf 5848  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860
This theorem is referenced by:  2fvidf1od  6513  2fvidinvd  6514
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