Theorem fcomptss 45219

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 6664	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
5		fcompt 7061	. 2 ⊢ ((𝐺:𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))
6	1, 4, 5	syl2anc 584	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑥))))

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3900   ↦ cmpt 5170   ∘ ccom 5618  ⟶wf 6473  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485

This theorem is referenced by:  ovolval2lem  46660  ovolval5lem2  46670