Theorem fvmptss2 6967

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 2821	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3		fvmptn.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmpt 6198	. . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	2, 4	elrab2 3649	. . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V))
6	1, 3	fvmptg 6939	. . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶)
7		eqimss 3992	. . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)
8	6, 7	syl 17	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶)
9	5, 8	sylbi 217	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
10		ndmfv 6866	. . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅)
11		0ss 4352	. . 3 ⊢ ∅ ⊆ 𝐶
12	10, 11	eqsstrdi 3978	. 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 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  ∅c0 4285   ↦ cmpt 5179  dom cdm 5624  ‘cfv 6492

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  cvmsi  35459