Theorem feqresmptf 45586

Description: Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
feqresmptf.1	⊢ Ⅎ𝑥𝐹
feqresmptf.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
feqresmptf.3	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
feqresmptf	⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem feqresmptf

Step	Hyp	Ref	Expression
1		nfcv 2899	. . 3 ⊢ Ⅎ𝑥𝐶
2		feqresmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2, 1	nfres 5948	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶)
4		feqresmptf.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		feqresmptf.3	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	4, 5	fssresd 6709	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
7	1, 3, 6	feqmptdf 6912	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
8		fvres 6861	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
9	8	mpteq2ia 5195	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
10	7, 9	eqtrdi 2788	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1542  Ⅎwnfc 2884   ⊆ wss 3903   ↦ cmpt 5181   ↾ cres 5634  ⟶wf 6496  ‘cfv 6500

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508

This theorem is referenced by: (None)