Theorem fvmptss2 6882

Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

Ref	Expression
fvmptn.1	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
fvmptn.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmptss2	⊢ (𝐹‘𝐷) ⊆ 𝐶

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

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

Proof of Theorem fvmptss2

Step	Hyp	Ref	Expression
1		fvmptn.1	. . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
2	1	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3		fvmptn.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmpt 6132	. . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	2, 4	elrab2 3620	. . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V))
6	1, 3	fvmptg 6855	. . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶)
7		eqimss 3973	. . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)
8	6, 7	syl 17	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶)
9	5, 8	sylbi 216	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
10		ndmfv 6786	. . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅)
11		0ss 4327	. . 3 ⊢ ∅ ⊆ 𝐶
12	10, 11	eqsstrdi 3971	. 2 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
13	9, 12	pm2.61i 182	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253   ↦ cmpt 5153  dom cdm 5580  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  cvmsi  33127