Theorem fcomptss 45112

Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fcomptss.a	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcomptss.b	⊢ (𝜑 → 𝐵 ⊆ 𝐶)
fcomptss.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)

Assertion

Ref	Expression
fcomptss	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐹   𝑥,𝐺

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fcomptss

Step	Hyp	Ref	Expression
1		fcomptss.g	. 2 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
2		fcomptss.a	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3		fcomptss.b	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	fssd 6766	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
5		fcompt 7169	. 2 ⊢ ((𝐺:𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
6	1, 4, 5	syl2anc 583	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3976   ↦ cmpt 5249   ∘ ccom 5704  ⟶wf 6571  ‘cfv 6575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583

This theorem is referenced by:  ovolval2lem  46566  ovolval5lem2  46576