Theorem fcomptss 45305

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 6674	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
5		fcompt 7072	. 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 3897   ↦ cmpt 5174   ∘ ccom 5623  ⟶wf 6483  ‘cfv 6487

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495

This theorem is referenced by:  ovolval2lem  46746  ovolval5lem2  46756