Theorem mptimass 5116

Description: Image of a function in maps-to notation for a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
mptimass.1	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
mptimass	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

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

Proof of Theorem mptimass

Step	Hyp	Ref	Expression
1		mptima 5115	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
2		mptimass.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3		sseqin2 3442	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
4	2, 3	sylib 122	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = 𝐶)
5	4	mpteq1d 4197	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵))
6	5	rneqd 4988	. 2 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
7	1, 6	eqtrid 2279	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∩ cin 3212   ⊆ wss 3213   ↦ cmpt 4173  ran crn 4752   “ cima 4754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-mpt 4175  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764

This theorem is referenced by: (None)